Just like casual shoes, sizing is a primary determiner for the shoe you’ll purchase, but running shoes are also divided into four categories: neutral, motion control, stability, and minimalist. Together these categories represent what’s called pronation control, which explains how the shoe supports the way your foot rolls when it hits the ground (how it pronates). Choose your shoe based on the correct size, the pronation control you need, and any preferences you have in the way of cushioning. Keep in mind that shoe fit varies between brands (read some reviews for fit clues) and avoid buying a shoe based on looks alone.
Trail running shoes are running shoes designed with aggressive tread patterns for traction, highly durable materials to protect your foot, and they may or may not use waterproof or water-resistant materials. These shoes are for off-road running.
Road running shoes are for running on asphalt and concrete (roads and sidewalks). With less aggressive tread patterns and lightweight materials, these shoes help you fly across hard surfaces.
A lightweight, minimal construction sub-category of both running and trail shoes. Minimalist shoes keep things simple and allow your feet to flex and absorb impact naturally.