Multi core, or stranded wires are used in applications that require supple wires. In environments that include vibrations, or where freedom of motion is necessary, multi core wire is more suited than single core (mono core or massive wire) which may break when moving a lot.
Single core wire is better suited for developing circuits on a breadboard.
The size of wires is given in mm2 or AWG (American Wire Gauge; for conversion see for example Rapid tables AWG-> mm2). In the metric system, a higher number implies a thicker cable, where in the American system, a lower number is indicative of a thicker cable. As a guideline, use at least 1 mm2 for 10 Ampère current.
Thin wires usually have a higher resistance then thick wires, and thus, a lower current carrying capability.
Also pay attention to the isolation voltage of the wire. Low voltage wire should not be used for AC-line applications.