Your car's clutch is what engages and disengages the transmission in your car. In a manual transmission, you need to apply the clutch pedal to disengage it and shift to different gears. The clutch is a wear item, meaning it will eventually need to be replaced in your car, much like how you eventually need to replace brake pads and other such parts, no matter your overall driving habits. When your car is having problems with the transmission, it may be that the clutch needs replacing, or you may assume the problem is the clutch when it's really the transmission or another part that needs repair. Note a few troubleshooting tips to tell you if the problem is the clutch or something else under the hood.
Lack of response
If you shift gears and there is a distinct pause before your car moves, and the vehicle seems to then "clunk" into gear, this often means a problem with the transmission itself. Your car may be low on transmission fluid, or there may be a worn chain or gear that isn't easily catching that change in gears. However, if the lack of response means that you put your car into gear and the engine seems to rev at a high RPM, but you're not moving as fast as you think, this often means a problem with the clutch. It may not be disengaging as quickly as it should when you change gears.
Grinding or shaking
If your car shifts gears smoothly but shakes when you drive, this might be a problem with your tires being misaligned rather than anything related to the transmission or clutch. If the car grinds after you shift gears, this often means a chain or gear in the transmission is worn and is grinding against another part. However, if the car makes a grinding noise while you're shifting or seems to shake as you change gears, this often indicates that the clutch itself is worn out and is not engaging or disengaging properly.
Won't shift into gear
Low transmission fluid can cause sticky gears, or the fluid in the transmission might be corroded and need replacing altogether. The transmission pan may have a leak or your car may simply need a fluid flush. However, if the fluid is not the problem, it's time to look at the clutch. If the teeth of the clutch are worn down or it's failing to disengage as it should, your car won't shift into any gear.