Bikes that only have front suspension are often referred to as “hardtails”. This means that the frame is rigid, and there is a suspension fork on the front of the bike which will hold the front wheel on the ground and provide some relief from the roughness of the road.
Full suspension mountain bikes are bikes that have suspension built into the rear of the frame in addition to a suspension fork in the front of the bike. The advantages of full suspension are that more of the roughness of trails is taken out and the rear suspension helps to keep the back wheel planted when going over rough terrain, giving the rider more control.
Some bikes come with neither suspension in the rear or the front. These bikes are considered fully “rigid”, meaning they have no suspension. The advantage of a rigid bike is that no power is lost during suspension modulation and so power transfer is much more efficient.