Having now completed one week of daily recitations of Psalm 27, I feel that it is now appropriate to offer some of my thoughts about the Psalm we will continue saying through the conclusion of Shemini Atzeret. The text with translation can be found here at Mechon Mamre. By the way, Mechon Mamre is a fabulous resource for your own learning.
The psalm seems to be divided into three sections. The first, verses one through three, the second verses four through six, and the third verses seven through thirteen. Perhaps not coincidentally JPS makes the same demarcations.
I read the first section as an affirmation of faith. It strikes me as similar to the opening bracha of the amidah where we invoke the names of our ancestors. Almost as if to say, no matter what happens we still have a connection to those people, and I, regardless of circumstance will maintain a connection to God. If you'd prefer a slightly more flippant take, the author is "buttering-up" God. Faith is being affirmed even at a time when individuals could have drifted from God's presence and faith is being recalled at a time when actions might not dictate recognition of God's existence. The reaffirmation serves as the first step to re-enter the abode of God.
Section two switches to a mode of petition and modesty. Our needs our constantly changing and we could request anything, but as the psalm says, "One thing I ask of the Lord..." shelter and protection. Even though I may have been wayward toward You or towards others, I seek Your shelter. Perhaps here we see a connection to Sukkot, which also explains why the Psalm is no longer said once we have completed our week in the sukkah. One final observation on this section: the root S'T'R appears twice. We are asking to be hidden and protected. Interestingly enough, the fifth sin found in the long vidui (confession) of Yom Kippur asks for forgiveness for sins committed in public or in sater (private), the same S'T'R root from our psalm. God knows the hidden sins along with the public ones, allows us to return after transgression, and provides protection. That's t'shuva.
If you believe as I do, that sin leads to sin, which eventually distances an individual from God and Godliness, then the third section flows with tremendous power. During the time of self-evaluation, while looking to grow in our relationships with each other and God we cannot help but ask to be heard and accepted by God. When I recite the text, my thoughts are usually directed to the idea that I do not deserve yet another chance. I am deliberately avoiding the temptation to over-dramatize, but the dramatic feelings do indeed percolate whilst reciting this third section. Yom Kippur's shma koleinu (hear our voice), which bears some similarities to the psalm, continues to be very personally moving, and in fact it is this third section, in conjunction with shma koleinu, that spurs me to continue delving deeper into myself.
I have put quite a bit out there today, but please feel free to comment and offer your own take.