Celebration is intrinsic to human nature. As God's image bearers we are compelled to celebrate—just as He did when He called His creation good, just as He did when He exalted His Son on top of a mountain, just as the angels and elders do around His throne. God even commanded celebrations in His name, providing the Jews specific instructions for feasts and holidays.
With the coming of Christ things changed a bit. God's people were no longer required to hold specific holy days above all others, because that was a feature of the Law that Christ fulfilled in our place and freed us from. The apostle Paul summarizes in Romans 14:5: One person esteems one day as better than another, while another esteems all days alike. Each one should be fully convinced in his own mind.
That's not to say there aren't good reasons to observe certain holidays, especially those memorializing key events or elements of the Christian faith. We look to the Sabbath, or Lord's Day, as our day of worship, and the Christian calendar is a good guide to those days traditionally observed by the Church, with special emphasis on Advent and Christmas, Easter and Lent, and Pentecost. If you're so inclined, there are a bajillion saint days in honor of famous and lesser-known champions of the faith.
Many will argue that Christians should celebrate none of these holidays. The point isn't that the events commemorated aren't central to the Christian faith, but that their formalized celebration by the Church was simply a "christianization" of similar pagan holidays in the Roman and Near Eastern worlds. For instance, Christmas as observed on December 25th is really a Christian version of the Roman Saturnalia, a festival devoted to the sun; Easter is named after the Babylonian fertility goddess, and celebrates her union with the god Tammuz.
The similarities between the Christian holidays and their pagan counterparts are indeed fascinating. However, we take this as affirmation of God's sovereignty, not as a reason to avoid holiday celebrations. Through His death and resurrection, Christ redeemed all things and reclaimed them in His name.
"All things" includes holidays. Indeed, what Christ has done and what He gives us should be celebrated every day of the year, but to choose specific days for observance can help re-center our all-too-often selfish existences on Him. We do need to be careful not to simply accept worldly standards for our holidays (the commercialization of Christmas, the nonsense Easter bunnies, etc.), but we need to be equally sure we aren't refraining from genuine celebration just because those standards exist. God has given us all cause for joy, and it's our duty and pleasure to thank Him.