Why does it matter if Voldemort uses your wand?

I just saw HP Deathly Hallows part 1. I'm not familiar with the lore. One thing I found confusing was, in the way beginning, when Voldemort is asking for people's wands to use so he can kill Harry, no one wants to give him theirs. Why is that? What's the big deal about a wand? Later in the movie, Harry goes through like three of them no problem.