Generally, there are only 2 or 3 parts to turntables that need attention, depending on the type you have. These parts are: the stylus or needle, the cartridge and in some turntables - a belt.
The Stylus or "needle"
The stylus is the part that is changed most often - as the friction created will cause wear as it tracks in the groove on the record. The stylus is held by the cartridge, which is fastened to the arm. The stylus is changed usually by pulling straight out - in the opposite direction of the turntable's arm, some you have to pull the front down first then out. A replacement stylus must be compatible with the cartridge. Take note of the cartridge manufacturer's name and model. If purchasing in a store, bring along the old stylus. Replacing the stylus is simply a matter of slipping it back into the opening in the cartridge, obviously with the point down.
The cartridge is another component that can be changed (but shouldnt have to be unless you want to upgrade it). Make the selection based on which types fit the turntable manufacturer and model. The cartridge usually plugs into a jack secured to a headshell or turntable arm. The cartridge is another "pull out" / "slip in" arrangement similar to the stylus.
Tracking pressure may need to be adjusted when changing the cartridge. Cartridges have different weights, so the counter balance (at the opposite end of the arm) and skew adjustments (usually an adjustment at the rear - near but not on the arm) will need to be set. When tracking pressure is properly set up, the stylus will "sit" fully in the record's groove. The anti-skew setting helps to keep the stylus pressure equal in horizontal directions - this helps prevent the arm from skidding across the record if bumped.
One of the most overlooked parts is the belt (direct-drive units do not employ belts). It's not plainly visible, so it's easy to forget its there. Indications for belt change include: the inability to adjust the turntable speed to attain the correct pitch of the recording - or to sync the speed to an indicator located on the platter to the strobe, or uneven speed or pitch.
In order to access the belt on some turtables it will be required to get to the underside of the turntable. Depending on model, it may be accessible by removing a cover on the bottom, or removing screws that allow lifting the entire mechanism up and out from the cabinet. If you have DJ decks they are normally designed to be accessed very easily so look under the record mat and see if you can see the belt. You might be able to just unhook it and lift the turntable off which will allow you to change the belt quickly.
Once visible, remove the old belt and clean all the surfaces that the belt contacts, with alcohol and a clean rag. When dry, slip on a new belt designed to fit the turntable and reassemble. Never oil these surfaces.