If you sign up for twitter, your username is @woofmanjack. So if I want to talk to you, I'd type "Hey @woofmanjack, our endzones are freakin' awesome #hailstate". Everyone who followed me could see the tweet, but you'd get a little notification that I tweeted at you.
The "#hailstate" is called a hastag. You can click on any hashtag in the tweet and it will take you to another screen in which is shows every tweet with "#hailstate" in it.
So essentially, the @ is used in front of a user name and the # is used as a hastag. Hope that helps.