Animals often require a complex support and leverage system for improved mobility: muscles attach to and move a bendable, often multi-jointed skeletal system.
In hydrostatic systems, resistance is provided by pressurized fluid chambers; squeezing the chambers causes them to distort. Often the muscles can squeeze in perpendicular directions: if a cylindrical space is contracted by longitudinal muscles, it gets wider, then contraction of circular muscles makes the cylinder longer. This is the mode of movement in annelids. Sometimes the muscles themselves are the fluid-filled chambers, as happens in cephalopod tentacles or vertebrate tongues. Echinoderms use a variation, a water vascular system, that delivers strong hydraulic power to extending and retracting tube feet.
Rigid systems are exactly what that sound like - the resistance is delivered from hardened structures. There are two basic types, with some systems combining aspects of each.
Exoskeletons are a secreted exterior rigid system with the muscles on the inside. These are complex systems: a clam shell is not considered an exoskeleton. An exoskeleton is protective, though, like a shell. Exoskeletons are a primary feature of arthropods. Endoskeletons have the hard parts on the inside, with the muscles around them, so there is no protection from them. Because of the way power is relayed over joints, exoskeletons produce a lot of leverage; they are proportionately stronger than endoskeletons. You may have heard how strong spiders or ants are - this is just partly the difference in leverage. Small muscles are also proportionately stronger than large ones. Muscles attach to the skeleton through tendons, protein-based cables that integrate into the connective-tissue wrapping sheath of the muscle.
Since they cover the animal, exoskeletons tend to be much heavier than endoskeletons. Really hard exoskeletal materials tend to be found only in aquatic animals, where the water helps with support; land arthropods use chitin, a material that is light and very strong in small structures. Even a light material, however, limits the potential size in animals with exoskeletons - with larger size, the animals get too heavy for muscles to move. Endoskeletons do not really have this limitation - they can be made of a very hard material, calcium carbonate (with some collagen protein added for resistance to breakage), and support extremely large animals. It is common for bones to be hollow, since solid columns are heavier and not much better support than hollow ones. They can be especially light with air spaces inside, as is found in birds (and probably at least some dinosaurs). In swimming vertebrates, fish, a bony skeleton was too heavy; skeletons were made of protein-based cartilage, still found in sharks, rays, and skates (they also have a complex interaction of muscles with a dense layer under the skin), before the evolution of a gas-filled swim bladder whoae buoyancy could counteract the weight of a bony skeleton, as is found in the vast majority of modern fish. The notochord is a cartilage-like material, eventually replaced by a cartilage vertebral column.
Endoskeletons can grow fairly smoothly inside an animal, something exoskeletons cannot do. An exoskeleton, once hardened, can't really grow; they must be molted, shed when the animal inside reaches a critical point. The animal must loosen the bond between epidermis and exoskeleton and then get out of it, which does not always work; many arthropods die trapped in their skeletons (insects do a limited number of molts, usually 5 to 6). The soft and vulnerable animal must then add volume quickly (usually by swelling up with water and/or air) and secrete a new exoskeleton.
Rigid skeletons require bendable joints. In exoskeletons, this must be done with softer, less protective materials. In endoskeletons, the bones are coated with cartilage, synovial fluid, and sometimes bursae. The movement of joints is limited by how ligaments attach the bones together.