An initiator is someone who can force a fight to happen and give your team a significant advantage while doing so. The key here is that when an initiator starts a teamfight properly, the other team shouldn't have the opportunity to simply run away, usually because of long stuns. This gives your team an opening to close the distance and follow up on the initiation.
The three most common initiators right now in high tier play are Tidehunter, Sand King, and Earthshaker- in that order. All three have significant AoE stun and AoE damage and with a Blink Dagger (or more and more frequently even without one) can give your team an opening or change a fight in seconds.
Other less commonly used or less capable initiators are Enigma, Puck, Vengeful Spirit, Tiny, Faceless Void, Stormspirit, Axe, Sven, Slardar, Warlock, and Clockwerk. Many of these are better as gankers because they specialize in killing a single target rather than dealing AoE, but will transition into initiators later in the game because of their ability to separate a key opponent and with a coordinated team eliminate him before any spells can be cast.