Generally speaking, there are two primary types of failure with very different symptoms, both of which have been mentioned above. As already noted above, hydraulic issues can also lead to a clutch that will not disengage properly, but I'm focusing on the clutch itself.
When the clutch disc friction material wears out (very similar to brake pads) the clutch will slip, preventing the car from accelerating in proportion to engine RPM. This is first noticeable under heavy loads and hard launches, but does not make it difficult to shift.
When the disk or pressure plate suffers mechanical failure (rather than just wearing out) it generally results in the inability to shift the car because the clutch does not disengage properly when the pedal is pressed. The disks have small horizontal springs to take up the shock of engagement and they sometimes break or jump out of their holder, making shifts very difficult or jamming the clutch so it won't disengage. The pressure plate is like a large spring-loaded clamp and the spring mechanism can fail. Although that can lead to a slipping clutch, it typically causes poor or no disengagement to the point that shifting is not possible, before the disk can be worn out.