Performance running shoes can come in wide variety of styles designed for different types of running styles. Generally speaking, there are 3 major categories that can help differentiate between the way people run. A runner who underpronates, meaning your feet roll outwards as you run and don't absorb shock naturally, is typically going to want a neutral and cushioned running shoe that also offers a decent amount of flexibility. This is typically a runner with a high arch in their foot. A runner who has normal pronation, or slightly overpronates, meaning your feet roll inward when you run, is going to want to look for moderate cushioning and stability that allows your foot work naturally. For runners that have flat feet, who typically overpronate, you will want to look for a motion-control, or stability shoe with relatively firm cushioning.
Once you've decided on which type of running shoe is right for you, there are many features to consider. Running shoes can be designed with many different uses in mind. Minimalist running shoes are designed to mimic the way your foot moves naturally. They are very lightweight and feature very little cushioning for those looking for a barefoot running feel. On the other end of the spectrum, trail running shoes typically feature extensive cushioning, and aggressive trade patterns designed for those that prefer a more challenging terrain.
Some common features to consider for running shoes include, breathable materials to keep your feet cool and comfortable, a minimal amount of stitching or seams in the toebox area to avoid irritation, anti-microbial and moisture wicking materials used on the inside of the running shoes, and a durable outsole.