The clutch system in your vehicle most likely works very similar to your brake system. In most cases, it may even use brake fluid as its working fluid. If you drive a classic car or truck you may have a mechanical clutch meaning it operates by a system of levers, pivot points and connecting rods to translate the movement of the clutch pedal into movement of your clutch pressure plate releasing the clutch. New cars use hydraulic clutch systems because they are much simpler to install, take up less space and do not require adjustment as your clutch wears down.
If you have replaced the clutch slave or master cylinder, the lines, or simply changed the fluid, you need to do an adequate job removing the air from the system before your clutch will work properly. This procedure is called "bleeding" your clutch system by most mechanics and is important because air in a hydraulic system will cause it to malfunction. As you press the clutch pedal, the master cylinder forces hydraulic fluid, in this case brake fluid, down through the line into the slave cylinder. The pressure of this fluid will cause the slave cylinder to activate, pushing your clutch fork and disengaging your clutch. If there is air in the system, that air bubble can compress. This compressing air bubble will absorb much of the fluid being sent from the master cylinder to the slave cylinder keeping the slave cylinder from actuating as far as it should, or even at all. If the slave cylinder does not travel as far as it should, your clutch will not fully disengage making it difficult to shift your transmission, especially into 1st gear from a stop.
To remove the air from your clutch system you need to push or pull the air down through the fluid line to the bleeder valve on the slave cylinder. To keep things clean you should attach a tube to the nipple on the bleeder valve. If you use a clear tube it can be easy to see when all the air has exited the system. The easiest way to bleed the clutch is to use a vacuum pump to pull the fluid and air out of the system while keeping the clutch fluid reservoir topped off. Continue pumping fluid from the bleeder valve until you no longer see air bubbles in the fluid. If you do not have a vacuum pump, you can do this manually with a friend.
With the bleeder valve closed, have your helper depress the clutch pedal as far as they can. Open the bleeder valve and allow the fluid to run out until it slows down, while your friend maintains pressure on the clutch pedal. When the fluid flow slows, while your Helper still has pressure on the clutch pedal, close the bleeder valve. Let the clutch pedal back up and repeat the process. Continue this process until no more air bubbles are seen exiting the bleeder valve. Your helper should feel the pedal (stiffen up) as you complete this process as air is easier to compress than the clutch plate release springs.
When you have completed this procedure make sure the bleeder valve is tightly closed, and go for a test drive!