A single coil guitar is a guitar that features a single coil pickup. A single coil pickup is a pickup (the part that picks up the sound on an electric guitar) that features only one set of coils in the guitar.
This creates a bright twangy tone that is useful for many forms of music, blues, funk, disco, pop etc. These pickups also cut through a band really effectively when distorted. Check out Jimi Hendrix for a great rock tone and Nile Rodgers for a fantastic disco/pop tone. Both musicians are using the same guitar model and it shows the versatility you can achieve from single coil pickups.
Two negatives of single coil pickups are that they can produce a loud humming sound (called 60 cycle hum) when at loud volumes or when using large amounts of overdrive, also some musicians claim to find them ‘too thin’ sounding are a not suitable for certain types of music.
Guitars that traditional feature single coil pickups are Fender Stratocasters. Fender Telecasters are Gibson Les Paul Juniors.
I would recommended getting a guitar with single coils for fans of these bands/artists or similar;
Jimi Hendrix
Pink Floyd/Dave Gilmour
Stevie Ray Vaughan
Blues
Soul/Motown
The Rolling Stones
Led Zeppelin/Jimmy Page
The Police/Andy Summers
Bruce Springsteen.