Modern door locks are usually part of self-contained knob-and-latch systems that are specifically designed to work together. This means that if the door lock breaks, you have to replace the whole knob-and-latch assembly to repair it. This isn't as daunting as it sounds, with the assemblies sold for easy do-it-yourself installation.
Remove the two screws from the latch plate on the front edge of the door (above and below the latch), using your screwdriver.
Remove the two machine screws on the collar of the interior doorknob. Pull both doorknobs straight out from either side of the door, removing them both.
Grip the latch, and pull it straight out from the front edge of the door, removing the latch assembly entirely.
Purchase the new latch assembly and knobs by taking the old unit to your hardware store, and finding the same size.
Slide the latch assembly into the smaller hole on the front edge of the door. Insert the two small screws through the latch plate, into the two screw holes that were left by the previous latch.
Insert the stems of the two doorknobs from your assembly through the knob holes and the back end of the latch assembly (which will be fully visible in the middle of the knob hole). Attach the doorknobs to one another with the two long machine screws that came with your kit. The knobs should turn the latch.