If you have been climbing with tennis shoes you are up for a nice surprise. Climbing shoes make gripping the rock a piece of cake. Climbing shoes are made of mainly leather and rubber that is designed to maximize friction on the rock.
The best way to choose a good shoe for you is to go to your local rock climbing store and try on as many as possible. After you know your size though, the internet has lots of great blowouts often allowing climbers to pickup shoes from 30-60 USD when they normally cost between 70 USD and 200 USD.
Do not expect climbing shoes to be the same size as your running shoe. They are often two sizes off. Rock climbing shoes should be tight but not painful (loose climbing shoes can limit your performance). Keep in mind that the leather on climbing shoes tends to expand after multiple uses.
DownturnedOnly shoes designed for performance are downturned (below left image is downturned; below right is not). Downturned is when the toe box is bent downwards. This increases the ability to stand on small holds and pockets. Disadvantages are that downturned shoes tend to be less comfortable than regular models and are not as optimal for smearing.
Full Heel Full heels tend to be better for heel hooking than slingshot models because they have more rubber connecting with the rock.
Slingshot Slingshot models tend to fit better than full heel models and are designed to force your heel toward the front of the shoe. This places a lot of power into your toes, which is better for micro edges.
LacingClimbing shoes are available with laces, velcro or as slip-on's. This is a matter of personal preference. Laces will create the tightest fit, but velcro / slip-on are easier to quickly remove between boulder problems.