Shapes vary based on the type of climbing. Shoes with an aggressively downturned shape offer enhanced foot dexterity, and are often preferred for hard bouldering and sport climbing. A flatter shape is ideal for technical face climbing and standing on small edges.
There are three types of closure systems: laces, hook-and-loop straps, and slippers. Laces adjust the entire length of the shoe for the most precise fit, hook-and-loop straps strike a balance of fit precision and easy on and off, and the stretchy upper of a slipper offers the fastest on and off, but sacrifices a little in the way of security.
For climbing shoes to work properly, they need to be fit tight. If you're new to climbing, you will likely experience some discomfort in the beginning. Sizing varies from brand to brand, but typically you'll want to choose a climbing shoe one to three sizes smaller than your street shoe size. Also consider stretch-unlined shoes will stretch up to a full size over time while lined shoes will stretch very little.