A flat cable is a multi-conductor cable which is formed by effectively laying a group of insulated wires parallel to each other, and bonding them into a flat entity, like a ribbon (hence the alternative name, ribbon cable).
The main advantage of flat cables is that connectors (of the so-called Berg, DuPont, or IDC form, with pins on .100" centers) can be connected to the cable in a simple press-on manner, avoiding the need to hand-solder each wire to pins in a connector, a tremendous saving in labour. They can also be connected in the middle of the cable, for multi-drop operation.
Another advantage of flat cables is that they are usually easier to route, especially when there are a large number of conductors; a round cable with the same number is usually harder to install, especially through limited clearances.
The technology which has almost entirely replaced older round cables, especially inside computers; the older round cables survive in external applications, since they are typically more robust.