A spell is an incantation (usually, but not always, Latin based) that makes use of the magical force of the wizarding universe to do things ordinary people cannot perform, like levitate objects, conjure fire, or stun a person.
Spells are most effective when the incantation is said with a wand, although they can also be carried out without the use of either wand or speech. Some of the more advanced wizards, such as Albus Dumbledore, Remus Lupin, and Lord Voldemort do not need wands to perform magic, for example.