Climbing endurance is the ability to hang on and climb for long periods of time. Endurance is not the same as strength. Strength equates with the ability to pull hard moves, such as long reaches, off small holds on an overhanging wall.
Endurance is long-term muscle strength and is a great base from which to build strength. Endurance is developed by high repetition and low resistance (lots of easy climbing).
In a rock gym, this means nonstop climbing generally using big handholds. Technical difficulty should be low and finger fatigue from crimping small holds should not be the limiting factor. The goal is the deep tissue all-out pump created by high mileage climbing.
If you are alone and bouldering, then traversing about and linking walls together without resting on the ground is a great endurance exercise. Shoot for 30 minutes of continuous climbing. Practice good technique; hang straight on your arms, use your feet, chalk up frequently, and milk rest positions and stems.
Climb up, down, left, and right, and be creative about different routes and moves. Start with lower-angle walls and move onto overhanging terrain as endurance builds. Remember to try to use large handholds that naturally accommodate an open grip. Too much crimping on small holds leads to sore fingers, not pumped arms.
Think in terms of laps to practice endurance climbing on a top rope or lead. Pick a difficulty level well within your limit. Remember that the goal is to climb continuously for several laps, so you do not want to be limited by technical difficulty. Try for five laps in a row. On a top rope, for example, climb to the top, have your belayer lower you down, and without touching the ground, start back up again. Repeat until pumped.