Our Lord entered Jerusalem riding a donkey. This is the fulfillment of prophecy. (Psalm 118:26; Zechariah 9:9) The people waved palm branches and laid them on the path before Him, crying out "Hosanna in the highest!" But in five short days, the crowds would shout "Crucify Him!"
Palm Sunday reminds us that Christ is God, worthy of our praise and worship. It also reminds us how fickle and sinful our human nature is. How quickly we can go from praising God to forsaking Him. Palm Sunday begins Holy Week, the week when we remember Our Lord's betrayal, mocking, denial by St. Peter, suffering, crucifixion, and death. Palm Sunday reminds us that although we praise Christ, we are nevertheless responsible for His suffering and death. Our sins crucified Him. Holy Week is the culmination of Great Lent, when we strive to purify ourselves and reflect continually on the sacrifice Our Lord suffered because of His love for us. The Palm branches remind us to always praise Him, not only on certain occasions, but in every moment and every action of our lives.
I would also add that the palm branches do not bring us "good luck." Our Faith is not superstition. The blessed palm branches are sanctified and holy, and thus they will anoint our home with the spiritual power that all sacred things possess (such as icons, crosses, etc.)
OK, that's my understanding about the meaning of Palm Sunday. I hope I am not too off base. I'm sure others can correct me if I have made any errors.