Five big factors to consider:
Road-running shoes are generally light, flexible and even cushioning with flatter and smoother soles.
Trail-running shoes have bigger lugs to grip on unstable terrain, stronger underfoot and generally stiffer to handle uneven surfaces.
Cross-training shoes are thiner and simpler, to be used indoors for workouts and balance activities.
The two main aspects are firmness and thickness of the foam between your feet and the ground. Cushioning can vary among maximalists (>30mm), moderate, minimalists and barefoot shoes (<5mm).
In all cases, one factor to observe is the Heel to Toe Drop, or simply “Drop”. Cushioning and drop are theoretically independent, but closely related. All cushioning styles have multiple drop options, except perhaps for barefoot shoes which generally have zero drop as well.
Depending on your usual foot strike type (neutral, overpronation or supination) you might want to prefer shoes that compound into a neutral gait.
To determine your pronation you can either consult with a specialist, or observe the wear patterns on an old running shoe. Not all shoes will be advertised as ideal for some specific pronation, but this is a common topic on footwear reviews.
Try the shoe with your usual gear (socks, insoles if any). Even better to test them at the end of a busy day, because the feet will swell as they would in a run.
A thumbnail of space in the front, snug on the sides and moves without rubbing. Use a Runner’s Loop to lock your heels. Each brand has a slightly different fit, so a lot of experimentation is need.
Relevant when buying used shoes, but also to know when it’s time to buy a new one. A good pair of running shoes should last from 500km to 800km. Keep an eye for the sole if it’s worn.