How can a star burn with no oxygen?
Can a fire burn in a room with no oxygen?
A fire cannot burn without oxygen. You can show this for yourself, in fact: if you light a small candle and then put a clear glass upside-down over that candle (without touching the flame), you can watch the flame slowly extinguish as it uses up all of the oxygen that you have trapped around it with the glass.
What if in that same room, with no oxygen, is it possible for a hydrogen reaction fire to start, like the sun?
The way the Sun "burns" fuel is completely different from the way a fire on Earth burns (the term "burning" is a bit misleading when used to talk about stars). The Sun gets its energy by smashing small light elements together to make heavier elements; most of a star's life is spent smashing hydrogen atoms together to make helium. The burning that a star does, then, is a nuclear reaction, and not a chemical one like the fires on Earth (when a candle burns, the atoms themselves remain unchanged: just the molecules are affected).
If not, how did the sun start to burn without oxygen?
So, the Sun can "burn" hydrogen to helium without the need for oxygen. It should be noted that in the presence of carbon, nitrogen and oxygen, stars heavier than the Sun may burn hydrogen to helium by using the C, N and O as catalysts. Even in these stars, however, an absence of oxygen does not prevent nuclear burning.
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