A missile is, broadly, a projectile, that is, something thrown or otherwise propelled.
Missiles can range from a rock thrown from a slingshot through a crossbow bolt to a Minuteman III intercontinental ballistic missile with multiple nuclear warheads. The largest missiles currently deployed represent the most destructive weapons ever made.
Rocket-powered missiles are known simply as rockets.
Missiles are often used in warfare as a means of delivering destructive force (usually a warhead) upon a target. Sometimes missiles are used to deliver other payloads. For example in the Gulf War cruise missiles were used to deliver reels of carbon filament to electricity stations and switches, effectively disabling them by forming short circuits.
Missiles which spend most of their trajectory in unpowered flight, and which don't use aerodynamics to alter their course, are known as ballistic missiles (because their motion is governed by the laws of ballistics). These are in contrast to cruise missiles which spend most of their trajectory in powered flight.
A guidance system takes data from the missile's tracking system and flight system and computes a flight path for the missile designed to intercept the target. It produces commands for the flight system.
The flight system causes the missile to maneuver. There are two main systems: vectored thrust (for missiles that are powered throughout the guidance phase of their flight) and aerodynamic maneuver (wings, fins, etc).
There are some similarities between guided missiles and guided bombs. A guided bomb is unpowered and dropped from an aircraft, and uses aerodynamic fins for maneuver.