The main thrust of this pericope is that the disciple's piety, alms giving, and prayer are not to be like that of the hypocrites who do their righteous deeds to be seen and admired by others.   The disciples prayer is also to be different from that of the Gentiles. 

The term piety is a good translation, but is a translation of the Greek for righteousness (dikaiosunh - dikaiosune)  See the discussion on righteousness under Mat 5:20 (Click on righteousness - Mat 5:20).
We understand piety - righteousness here as a right conduct which has its roots in the right relationship with God and an attitude of commitment to the will of God.

Although disciples are to let their lives shine as light (Mat 5:14) they are not to do so in order to be seen and admired by men.  Here, although piety is a reflection of ones relationship with God and his will, it is not to be performed in order to be seen by men.  The reward of a public display of deeds is the public acclamation of men.  The reward of righteous piety is the reward from the Father in heaven.

Matthew appropriately connects the piety with the practice of alms giving (Mat 6:2).  the word used for alms giving is the same as that for piety, namely, righteousness (dikaiosunh - dikaiosune).  The practice of giving money to the poor was common among the Jews.  We have already noticed Jesus' concern for the poor (Mat 5:3).

It is apparent that Jesus was reacting to the public "performance" of piety practiced by many of the Jewish leaders of his time.
That the matter was a Jewish problem and one of considerable concern to Jesus can be seen in the language used  "as the hypocrites do in the synagogues...that they may be seen by men...."
The term upocriths - hupocrites literally means an actor, one who performs in front of others!
Apparently Jesus took exception to the "righteousness" of the scribes and Pharisees for in Mat 23 he refers to them as hypocrites.  He saw in the Jewish leaders of the day a insincerity that was not befitting of the kingdom.

The expression "do not let your left hand know what your right hand is doing..." is an Hebrew idiom indicating the discretion needed in the conduct of benevolent works.

Jesus repeats his comment regarding reward.  Public demonstration receives human reward.  Discreet benevolent works (Jesus' expression was alms given "in secret") are noticed by the Father who privately takes care of reward! 

Jesus again observes that those who pray to be seen by others have their human award in the acclamation of others, but one should pray "in secret" or unobtrusively to be seen by God who then rewards "in secret."

In regard to the Gentiles, apparently the "empty phrases" (battalogein - batalogein - babble) related to attempts to manipulate God by the repetition of words in a "magic" formulae, not merely the repetition of words.

The strong emphatic construction (in the Greek) with which Jesus begins his warning in Mat 6:5 ("But when YOU pray...") indicates Jesus' deep concern that the disciple's prayers be different.  
From his instruction and his own religious lifestyle, prayer should be an important aspect of the disciple's righteousness.

What follows is commonly referred to as The Lord's Prayer.  Some take exception to this nomenclature and prefer to call this The Model Prayer.  Such "splitting of hairs" is not necessary!  Certainly this is not a prayer prayed by Jesus and is intended to be a model prayer, but in contrast to model prayers taught by others (John the Baptist to his disciples) this is Jesus' model prayer, so can legitimately be referred to as The Lord's Prayer!

Luke records that in this prayer, or at least on a similar occasion, Jesus' disciples asked him to teach them to pray "as John taught his disciples..." (Lk 11:1).

Jesus' words in Mat 6:9, "pray then like this" (outws - houtos - like - in this manner) indicate that this was to be a model prayer.

It is vital in understanding Jesus' model prayer that we remember that it was given in the context of Judaism, for Jesus was a Jew, the disciples were Jews, Matthew and his community were Jews.  We must, therefore, interpret it in a Jewish context.

From what we can learn of Judaism in Jesus' day His model prayer was shaped by the prayer tradition and pattern of the synagogue.
There are a number of similarities between Jesus' prayer and the regular synagogue prayer commonly known as the Qaddish.  (Students interested in the Qaddish can click here on Qaddish and go to a version of this prayer.)

Notice the opening thoughts of the Qaddish, and the similarities to The Lord's Prayer:

An important factor in understanding synagogue concerns, and especially the Qaddish, that we recognize that their focus or primary concern was a redemptive theme, and not concern for mundane (earthly) matters. 

That this prayer had an impact n early Christianity is seen in that it is found in a document commonly known as The Didache.   The Didache is an abbreviated form of the full title of the document, The Teaching of the Twelve Apostles.  This document is one of the earliest documents of Christianity dating from approximately 115 AD.  This interesting document contains teaching and church instruction relating to the early church.  The copy of the Lord's Prayer in The Didache (ch. 8:2) is very close to the Matthean form of the prayer.

The Lord's Prayer:
The first thing we should note is the remarkable structure of the opening words of this prayer. The structure has been lost in many English translations, but is important to our understand what Jesus is calling for in this prayer.
note the parallelism of Mat 6:9,10.

We have here an interesting structure!  The words in bold in the Greek text are three parallel imperative forms of  verbs
    agiasqhtw - hagiastheto - "let it be holy.."
    elqetw - eltheto  "let it come..."
    genhqhtw - genetheto - "let it be done..."

Translated literally these verses read:

These three parallel imperatival clauses are all modified by the final clause of the petition, namely, "on earth as it is in heaven."  The Greek text has this in reverse order "as it is in heaven so on earth" emphasizing "so on earth."

What this means is that disciples are to pray that:

This prayer is not a premillennial prayer that God's kingdom would come sometime in the near future, but rather a prayer that God's kingdom would be on earth just as it is in heaven.
This is brought out by the nature of the other two parallel statements:

We mention again that the concept of kingdom in the bible is an emphasis on God's reign.  This prayer is that God would reign on earth just as he reigns in heaven.  A prayer that should be on every concerned disciples lips.

The second petition (6:11) of the Lord's Prayer on the surface looks simple, but in reality poses or suggests several possibilities.  Just what does Jesus mean by "Give us this day our daily bread"?
The Greek is not easy to translate!
RSV - Give us this day our daily bread
NIV - Give us today our daily bread
NRSV - Give us this day our daily bread, but includes in a footnote an alternate reading  Our bread for tomorrow

Literally, the Greek reads "The bread of ours for tomorrow give us today."
Hagner translates this "give us today the eschatological bread that will be ours in the future."
Hagner sees in the term epiousa (epiousia)  a reference to the future.  This is possible, for the word epiousia can mean a wide range of meanings, including, "for our need, for tomorrow, for necessary existence, for today, for tomorrow, for the future, and, next"!
The student may rightly ask, how can it mean so many things?
Words take on meaning not simply from the etymology (root meaning), but from their context, or how they may be used in a particular setting.
Epiousia is a compound word built on tow words, epi and ousia.  Epi means among several possibilities, upon, on at, by, for, et al.  Ousia likewise among several things means substance.  Adding these two words together offers several choices as seen above.
The problem is choosing which fits the context better!

Since the Qaddish  had a strong eschatological essence and was more concerned for redemption, Hagner has chosen future bread of redemption, or eschatological bread.  Hagner has set the bread in the context of the eschatological (end of the world) banquet which has a significant role in the Jewish and Christian psyche and theology.  The eschatological banquet was the hope (promise) that Jews and Christians have that at the end of the world all of God's children will sit at his table and eat (benefit from) his spiritual food that will sustain them for eternity.  This is a strong and significant theological theme of New Testament theology.

We are inclined to adopt the interpretation given by Hagner, that Jesus is not simply concerned for the mundane physical food (although this would fit well into the prayer for daily sustenance), but for the spiritual food that sustains us for the future.

If this is a simple prayer for physical food, then it is the only emphasis on the physical that one finds in The Lord's Prayer.  All the other petitions are for spiritual concerns.

The next petition (6:12) "forgive us our debts as we forgive our debtors" is a prayer for the forgiveness of sins.
Ofeilhmata - opheilemata literally means "shortcomings" but was a common word used by the Rabbinic literature for "sins".
Again we se the emphasis that we should not expect more than we are willing to give!  In Mat 5:7 we saw that it would be the merciful who could expect mercy.  This same theme is repeated in reverse in Eph 4:32, and is discussed in detail in mat 18:21-35.

The point in this petition is, however, that disciples should pray to God on a regular basis for the forgiveness of their sins, but also that disciples must be forgiving like their Father in heaven.

The final petition (6:13) relates to temptation, "and lead us not onto temptation..." 
This opening clause may cause some concern for those of us in a modern western setting: a plea that God would not lead his children into temptation!
First, the word peirasmos - peirasmos - can mean both temptation and testing.
James 1:13 informs us that God does not tempt his children.  James warns his readers

However, from what we have already learned from Jesus' temptation, God does allow his children to be tested.  This is one of the secondary lessons we learn from the OT Book of Job.
In the translation of peirasmos in 6:13 we prefer the meaning of testing.
However, we will not understand this petition until we see it in parallel with the second clause of this petition, "But deliver us from evil."
Perhaps it is better to translate evil as The Evil One.  The noun structure tou ponhrou - tou ponerou can be either masculine or neuter gender.  If we take it as a neuter then we would understand it as evil.  If we take it as masculine it could be The Evil One.  The difference is not worth debating!

The word deliver ruomai - ruomai can mean save, rescue, deliver.

This petition is a plea to God that He deliver us, rescue us, guard us, and protect us when we face testing and trials.

The final verses of this pericope (6:14, 15) repeat the oft stated warning and encouragement to forgive one another as the father forgives the disciple.  However, here the statement is in a conditional form "if you forgive...your Father will forgive your trespasses."
Paraptwmata - paraptomata is the standard word for trespass, meaning in the biblical sense of falling from the right way, namely, God's way.

Fasting

The whole subject of fasting is interesting.
The first point we should make is that this is a Jewish practice that is more sociological than theological in the sense that fasting was not commanded but practiced.
On numerous occasions we read of people in Scripture fasting, but nowhere does it seem that it was commanded!
We learn from the Rabbinic tradition that the Jews engaged in both public and private fasts.
Fasting was engaged in on festive and ceremonial occasions and solemn occasions of grief and mourning.
The Jews fasted on the Day of Atonement, on the New Year, on days of mourning and on the anniversaries of notable calamities
The Rabbinic tradition informs us that stricter Jews also fasted regularly on special days such Monday and Thursdays.

From what we learn in the Gospels, Jesus was neither proposing that disciples fast nor condemning the practice of fasting.  He spoke out against hypocritical fasting which had as its goal "to be seen by men."

From the Rabbinic tradition we learn that some fasting Jews "disfigured" their faces by nit washing, by not keeping their hair tidy,  and by smearing ashes on their faces.  It was this ostentation that Jesus spoke out against.

Anointing the head (with oil) and washing the face which Jesus suggested was in opposition to the public "disfiguration" of the face.

Today, fasting is not a regular practice among disciples.  The main reason for this is that fasting is not a sociological phenomenon.
This should not be interpreted that disciples should not fast!  If disciples, or a disciple, chooses to fast as an occasion of deep spiritual mourning or commitment, this is admirable and to be commended.  However, it should be a private matter and not made a spectacle. 
When disciples "demand" that others joint them in a fast, or when the impression is left that only disciples who fast are true or spiritual disciples, or when the impression is left that fasting is a sign of spiritual maturity, the we, lie Jesus, should speak out against the abuse of a fine practice.

We might suggest that it would be admirable and edifying if disciples did declare times of private commitment, reflection on Scripture, meditation, and prayer.  But this should be conducted with due care that one's purpose does not come into conflict with the freedom disciples have in Christ.