A basic attack is a term for a single target attack ability, which has no Cooldown associate with it. Many strong attack-related abilities (such as Swarm and Split) only trigger off of basic attacks, and not attacks in general. It is easy to tell if an attack is basic or not -- if it is a basic attack, it will say "basic attack" in it's description.

Most champions have a basic attack of one or more damage types, such as Attack: Frost

If the ability description simply says "attack", and not "basic attack", then it is not a basic attack .  Many special attacks, such as Stab, the attacks made by Manic, and the attack from Heavy Crossbow are not basic attacks, even though they appear to be similar.

Also see Basic Attack Sources for basic attack sources.