We are a commune of inquiring, skeptical, politically centrist, capitalist, anglophile, traditionalist New England Yankee humans, humanoids, and animals with many interests beyond and above politics. Each of us has had a high-school education (or GED), but all had ADD so didn't pay attention very well, especially the dogs. Each one of us does "try my best to be just like I am," and none of us enjoys working for others, including for Maggie, from whom we receive neither a nickel nor a dime. Freedom from nags, cranks, government, do-gooders, control-freaks and idiots is all that we ask for.
In this case your lectionary omits not just the conventional opening verse - "for the choirmaster, a song of David" - it also leaves off the important second verse:
"When Nathan the prophet came unto him, after he had slept with Bathsheba."
So this is one of the central prayers of David's personal life - similar to the Psalms surrounding his conflict with Abshalom.
2) Deal graciously with me... according to your mercy.
"Deal graciously with me" is literally "pardon me".
3) Scrub me clean from my transgression, and purify me from my mistake.
The word for "wash" is the same one used for beating laundry, and the word translated as "clean" is related to the word used when the Israelites walk on "dry land" in the Red Sea - a landscape "scrubbed clean".
I've translated as "mistake" the word used for "sin" - it literally means "missing the mark" and is used in modern Hebrew sports reporting to describe a missed goal.
4) "blameless" is better translated as "merited" - G-d's case against me has merit.
5) The words for guilt and sin are the same words used in verse 2 above, creating an echo... the word for "conceive" here literally means "conceive" - to tell or think of something - and is unusual.
6) "Inward being" is literally "the plastered over places" - a single word. We've previously discussed this image for human self-justification... "in my secret heart" is simply "the sealed up/silent"
7) "Purge me with hyssop" refers to the purification rituals of Leviticus - the priest uses a hyssop branch to sprinkle water or blood on the ritually impure... Hyssop is a low-growing shrub, a symbol of modesty and humility.
"Wash me" is again the more vigorous "scrub/launder" and the same word is used to describe the thorough washing of clothes that accompanies the sprinkling ritual.
9) Again, the same 2 words as above for mistake and sin.
10) "New and right" is simply "steadfast". It's a word meaning both "established" - like the foundation of a building - and "correct".
11) do not cast me from before you...
12) "Willing spirit" is really a "generous spirit"
Again your lectionary cuts it short, omitting some famous verses, like:
The sacrifices (beloved) of God are a broken spirit;
a broken and a contrite heart, O God, Thou wilt not despise.