It's a way to make computers understand what you're talking about. If you're chiming in on last night's Jays game, you add #Jays, #BlueJays, or #TorontoBlueJays to your message (no spaces allowed, that's why words are mushed together). For example, you would tweet:
"What the hell was up with Bautista last night? He was phoning it in. #Jays"
That way, Twitter's massive army of computers can show your tweet to everybody else who's talking about the game.