Dora Bas Rivka Silver O'H
The musical note for this Psalm says "on the ayeles hashachar," or "morning star." Presumably, this is the name of an instrument, although some commentators see it as a prophetic allusion to Esther and the Psalm a reference to Purim. (Many Psalms contain prophetic allusions or references to future events. For example, David speaks frequently of the Temple, though he was not permitted by G-d to build it himself. Rather, David's son Solomon was to build it after David's death.)
David starts by crying out, "My G-d, my G-d, why have you forsaken me?" Why has He turned His back to our prayers? We call out day and night and receive no reply. G-d is the Holy One and His (metaphorical) throne is (metaphorically) perched on the praises of His people. Our ancestors in Egypt put their faith in G-d and He saved them. We are insignificant compared to them and looked down upon by all the other nations. Rely on G-d - He will save us!
Ever since birth, He has been our G-d and we ask that He not distance Himself now, in our time of trouble. We are surrounded by threatening enemies and we simply melt and collapse from fear. They are like a lion at our hands and feet, gloating over our misfortunes. They divide our garments by casting lots. Please, G-d, do not be far from us!
When G-d saves us, we will praise Him to the other Abrahamic nations - Rome (descended from Esau) and the Arab nations (descended from Ishmael). Let all who are in awe of G-d praise Him because He cares for the poor and downtrodden when they call upon Him. I will praise G-d publicly and fulfill my vows to Him. Those who seek G-d will praise Him; may He be in their hearts forever!
When the nations of the world see all this goodness, they will turn to G-d and worship only Him. Even the already-prosperous ones will recognize Him, though there are some crimes He will not overlook. G-d will be eternally praised by His people, who will share His greatness with future generations.
This Psalm is actually written in the first-person singular ("I"), though I have paraphrased it here in the first-person plural ("we"), since it is unclear to whom to ascribe the statements. Is it David speaking autobiographically? Is it David speaking prophetically in the person of Esther? (For example, the part about G-d being our G-d since birth would refer to Esther being orphaned at birth.)
Some Christians see this Psalm as a Messianic prophecy, though this relies upon several mistranslations. For example, verse 17 clearly says, "My enemies are like a lion at my hands and feet," not "...have pierced my hands and feet." (In the Christian Bible, it's verse 16, as they do not count the introductory phrases as separate verses.) Similarly, the book of Mark ascribes verse 2 (to them, verse 1) to Jesus as his last words, although the word for "forsaken" here is azavtani, not sabachtani, as it says there. In Jewish thought, this Psalm is not a Messianic prophecy at all, although the last few verses, referring to the nations of the world praising G-d, could be seen as referring to the Messianic era.