All rockets generate thrust by igniting an mixture of fuel and 'oxidiser'. Liquid fuel rockets hold these two components in separate tanks, only allowing them to mix together in the combustion chamber at the base of the rocket. Liquid fuels are often difficult to handle, because the best fuels, liquid liquid hydrogen, and oxidants like liquid oxygen, need to be kept at very low temperatures to stop them boiling into gas. Solid fuel rockets, on the other hand, do not have this problem. They use different chemical fuels and oxidisers, which are mixed together from powder with a rubbery 'binding agent, to form cylinder-shaped 'fuel grains'.
Solid fuel is much easier to handle than liquid, and so is also cheaper.
It can even be more powerful than liquid fuel rockets. But there are major disadvantages as well - once ignited, by a spark at the base of the rocket, the fuel burns like a firework, ad will not stop until it is all used up. By contrast, valves can be used to vary the amounts of fuel and oxidiser being burnt by a liquid rocket, which allows the thrust to be changed throughout the 'burn'. The fuel can even be cut off completely and reignited at a later time. Solid fuel rockets can be designed with special features to change the thrust the rocket produces during its flight, but still don't have the flexibility of liquid rockets.