Closure

Lace

The only type of shoe to have a lacing system. This allows for a more customized fit as you can tighten some areas and loosen others. Lace-ups are generally designed for more comfort (flatter lasted and more symmetric) because they're intended to be kept on longer than an easily removed slipper or velcro shoe.

Slipper

There is no separate tongue and the top of the shoe is a stretchy material. Nice for bouldering because they're fast to put on and take off between burns. Some crack climbers also like them because it creates a smoother top surface with less bulk for jamming. Since the only adjustability is the stretchiness of the elastic material, it's extra important the shoe fits well from the start.

Velcro

Also fast to take off and put on, but with some extra adjustability. The velcro straps secure the foot which helps to keep the heel in it's place. The most popular shoe type thanks to their versatility.

Slipper + Velcro

A recent hybrid: When the top of the shoe is stretchy like a slipper (no tongue) and also includes a velcro strap. This allows a more customized fit than a traditional slipper as the velcro helps secured the heel and arch. These are usually very downturned, performance oriented shoes.