The term full-frame is actually a way to mimic the film size of erstwhile 35mm film. A full-frame camera thus has a sensor inside it which has a size of 36mm x 24mm.
There are sensors which are much smaller than that of full-frame, just as there are sensors which are larger than full-frame. A popular DSLR sensor size is APS-C.
Also known as Advanced Photo System type-C. Full frame systems being larger than APS-C systems, they have larger individual pixel sizes, if the resolution of both sensors is the same.
There are many advantages to using a full-frame sensor when compared to an APS-C camera. One of the advantages is that you can collect a lot of light compared to smaller sensors.
Additionally, the low light performance of these sensors is better. Finally, these sensors capture a much wider field of view.