Some groups of Christians fast during Advent, and some do not; while most Christians observe Christmas, the customs for the weeks preceding Christmas vary widely across denominations. Eastern Orthodox Christians fast during Advent in a way similar to their Lenten fast. At the other extreme, many Protestant Christian churches do not follow the seasons of the church year at all, so Advent, Epiphany, Lent, etc., have no importance to them. My Episcopal church is somewhere in between: We do observe the church year, and our worship services during Advent are characterized by solemnity and a sense of self-examination and preparation for the coming of Christ, but many (most?) Episcopalians do not fast during Advent in the way that the Eastern Orthodox do.
Because practices vary so widely, my advice is: Don't feel like you're required to fast during Advent. However, if you've observed a Lenten fast before and it was spiritually helpful to you, then consider fasting during Advent as well. Or, if not a fast, consider adding an Advent-themed reading or meditation to your daily prayer life during December.
I'm still working out what my own best observation of Advent should be. Here in the US, Advent is so completely at odds with the culture around us that it's hard to fast -- everybody's Christmas parties are during Advent, sigh. But I do try to spend some time each day in reading and contemplation on some Advent-related theme, and I find that helpful for me.
