Lay out the harness on the ground or hold it in front of you to ensure that there are no twists in the leg loops or the waist belt. At the front of the harness will be where there is a loop connecting the waist to the legs, or sometimes one in the middle of the strap that goes around your waist.  This is called the belay loop.

Step into the harness so that one foot is in each of the leg loops.  Make sure you also stepped through the waist belt.

Pull up the harness until the waist belt is above your hips.

Tighten the waist loop by pulling on the straps.  Newer harnesses automatically double back but many older versions need to be manually double backed.  Make sure this is done properly or else your harness may come undone while climbing.

To check to see if your harness is tight enough, try to pull it down over your hips.  If it stays on, you're good to go.

Tighten both leg loops the same way you did the waist strap.  Leg loops should be left slightly loose so that you can still move your legs.  A good rule is to be able to fit two fingers together between the leg strap and your leg. On some harnesses there are straps behind you leading from your waist to the leg straps. Tighten these until they're skin-tight.