Squeaking disc bicycle brakes are loud and annoying. They're usually a result of worn or contaminated brake pads or oil or other moisture on the brake rotor. Inspection, cleaning and replacement of the brake pads, along with cleaning the brake rotor can often alleviate the problem. With normal use, the new brake pads will silence the squealing and return your brakes to full power after a short break-in period.
Remove the bicycle wheel. Once the wheel is off, don't squeeze the brake lever. This is important! If you squeeze the brake lever, the disc brake pistons will try to squeeze the rotor and end up too close together when it's time to put the wheel back on. It also makes it a little harder to insert the new brake pads.
Remove brake pads. Undo the retaining pin using the Allen key if it's a threaded unit, or if a pin and circlip is used to hold them in place, remove them with the pliers. If the brake pads are shiny, use sandpaper to roughen the surface, and remove any debris embedded in the pad. If they are worn out (0.5mm or thinner) and there is very little pad material left, replace them with new pads. There are two kinds of brake pads: sintered and organic. Sintered pads last longer and perform better under wet conditions, but they are more prone to squeaking. You may choose to try out both to see which ones work best for you.
Clean the disc rotor with rubbing alcohol and a towel. Be careful of the sharp edges. Make sure the rotor is completely dry before you reinstall the wheel on the bike, to prevent contamination of the brake pads. When replacing pads, always use the new spring that comes with them, as the old one may not fit properly and the pads may rub on the rotor.
Slide the new brake pads into the caliper and reinsert the retaining pin. After you have replaced the brake pads, you can also "center" your caliper and rotor by loosening the mounting bolts that secure the brake caliper to the fork and applying the brakes. The brake unit and fork will centre themselves over the rotor.
Tighten the mounting bolts in the new position. Make sure all cable or brake line housings are seated properly at any stops along the length of the bike frame or fork.
If your bike uses cable disc brakes, keeping them working well is simply a matter of ensuring the cable is tight and runs smoothly in the cable housing. Hydraulic disc brakes are a little more complicated. They can require "bleeding" of the brake lines to ensure top performance. If you are unfamiliar with this process, it's best to have a bike shop mechanic take care of it, or take a class in bike maintenance before attempting to bleed the brakes.
Brake adjustments should be made only by experienced mechanics. If you aren't sure of your skills, have a professional bike mechanic look at your brakes.