Okay, I've had this question asked to me a
billion times in my philosophy class
Actually, I have a project to do in that class. All we have to do is "Define and prove your existence" ...but we can't do it in an essay

Damn creative ways! If anyone has any ideas for what I can do, feel free to help me out, ha ha
But yeah, back to your problem... I can share with you a few points I jotted down for my project...
Fist off, we have "Why it can't be a dream," since that's what my teacher always likes to claim. "It's all an illusion! We are actually sleeping, Matrix style, in some crazy pods!" So, my arguments where...
- We'd all have to be having the same dream, since we interact with each other. Highly unlikely.
- We can't think logically in dreams, feel pain, OR question our actions. We can in life.
- You can't have an illusion within an illusion; it's too complicated. Too much room for errors that would make us question the fake 'reality' we are living in. So pretty much, dreams wouldn't work.
And if all else fails, go with the good old point "Why does it matter?"
- If it is all an illusion, we will never know otherwise.
- As long as we believe it's reality, it becomes our own little reality.
- If we find out it is an illusion, then we just need to stive for our actual reality.
Some thing you might want to talk about if arguing the "Why does it matter?" thing is "The Allegory of the Cave" which was posed by Plato.
Here's the picture:
http://lssacademy.com/wp-content/uploads/2008/01/cave.jpg
It's pretty much a story about these prisoners, chained to some chairs in a cave. They've been there all their lives and don't know anything else exists. Behind them is a fire, and their captors walk in front of it with weird shapes (cat, tree, etc.). The shadows show up on the wall the prisoners see, so to them, that is life. That IS reality. One day, one of the guys breaks the chains and goes free. He wanders out of the cave, and sees the world.
Now, he would have to go back and tell everyone what he saw, but the thing is, is that the right thing to do? If he tells them, some might love him, saying "Thank you! Now I know there is more to life!" Others might hate him, saying, "What the heck man! I was fine when THIS was my reality! Now I'll be depressed all my life knowing what's really going on!"
I'm not sure if that story will help you any, but who knows, it might
But one thing you
SHOULD know is this:
"Cogito ergo sum" It's a Latin quote by René Descartes. It means
"I think, therefore I am" It pretty much means if we can question our existence and reality, then that is proof enough that we DO exist and it IS real.
...I hope that helps you, ha ha! And if anyone has any ideas for my project, please, feel free to share