There are two basic types of running shoes: road running shoes and trail running shoes. Most people are just going to need road running shoes, which are good for things like pavement and other packed surfaces with only mild irregularities. You only really need trail running shoes if you are going to be running in very irregular off-road situations with roots, rocks and other things getting in your way.
More important, though, is fit; so if you think you need a trail running shoe but can't find one that fits your foot properly, go with a road running. There is usually a wider selection of them.