Climbing boots, or shoes are possibly the most invaluable tool a climber has, without them many modern climbs would be nearly impossible.
Climbing shoes have thin rubber soles that allow climbers to feel the rock as they climb, exert some pressure from the toes to grip in pockets or on edges whilst maintaining the most possible friction with the wall.
Different types of shoe are favoured for different types of climbing. Shoes with good ankle support and a wider sole are better for longer climbs as they are more comfortable to wear (being wider) and are more rigid so are better for technical edging. A slipper type shoe allows for more flexibility so is good for bouldering and short sport routes, they are not as good at edges but are generally better at smearing.
A general rule of thumb when buying climbing shoes is to buy the smallest pair your feet can fit into, in affect making the shoes tighter which increases the support and sensitivity of the boot. Many climbers jump down one or two sizes, apparently putting on climbing shoes should bring tears to your eyes, and taking them off should make you weep with relief.