Atoms have north and south magnetic poles--just like the Earth. Although everything is made of atoms, most things don't behave magnetically because the poles of the atoms are not aligned--the poles point in all different directions. When something aligns the atomic poles in a substance, the substance becomes magnetic. Electricity is one of the things that can align the poles of atoms.
The archetype electromagnet is the crane-operated model that picks up automobiles and scrap metal by the ton. This model demonstrates one of the desirable features of the electromagnet--it becomes a magnet or not a magnet at the flip of a switch. The electric current running around an iron core aligns the iron atoms to make the iron core a magnet. A smaller application is the doorbell where an electromagnet moves a striker to hit the bell. Speakers are another application of electromagnets. A paper cone is attached to an electromagnet, which is controlled by a varying electric current. The singer sings, a matching electric current is generated, the electromagnet receives a rhythmic input and the paper cone vibrates to reproduce the singer's voice.
Motors use magnetic fields to rotate a shaft. As the electric current going to the motor varies--all generated currents do, it causes the rising and falling magnetic fields to push the core of the motor around. Motors are ubiquitous--at least a dozen are in your car, there is one in every appliance, there is one in your computer to turn the hard drive, and there is one in the automatic door at the supermarket.
When a tiny electromagnet is moving over an area on a magnetic data storage medium, it will leave a magnetized spot if the electromagnet is turned on and no magnetized spot if the electromagnet is turned off. Later a loop of wire is rapidly moved past the spot and the field from the magnetized spot will induce a tiny electric current. In this way information is read and recorded. Because the read/write device does not actually have to touch the medium to record by magnet field, the devices can move past each other very quickly and data can be read and recorded at tremendous speeds.
Magnetic levitation, or Maglev, applies a property of disk drives to electric trains. If a train can ride just above the rail, on a magnet field, there will be very little friction and it will be easy to move the train. Naturally, the train could then run very fast. This is how the Japanese bullet train--Shinkansen--works. Because the trains are powered through the rails, it is easy to build the rails in blocks that allow only one train at a time to be on a block.