A single coil pickup is a type of magnetic transducer, or pickup, for the electric guitar and bass. It electromagnetically converts the vibration of the strings to an electric signal.
As its name implies, the single coil pickup consists of only a single coil of wire which typically wrapped around a set of magnetic pole pieces. These pole pieces project a magnetic field which the guitar's wire strings disturb, thus inducing a small current. This current is then sent to an amplifier, which increases the current and sends it to a speaker which converts it into sound.
Several variations of the single coil pickup exist: Tele-style (which has distinct neck and bridge designs), Strat-style (often used in a set of three), P90s, Jazzmaster, lipsticks, etc. Each of these styles typically has a distinct size and mounting method, so when replacing single coil pickups you need to make sure you get the right type for your guitar.
Beyond the basic form factor, the sound of a pickup can also be changed by using a more powerful or different type of magnet, changing the number of wraps or the gauge of the wire used for the coil, changing the position or type of pole piece, etc. Due to all of the variables involved, a virtually limitless array of sounds can be achieved.
Single coil pickups are one of the two most popular designs, along with dual-coil or humbucking pickups. Humbucking pickups are, as their name implies, less susceptible to the hum that can be caused when the pickup coil encounters electromagnetic interference.