There are so many different health problems you can look for - but it's a lot better to pay a good equine vet to look at the horse. Sounds silly for a horse that isn't very expensive, but since keeping a horse is expensive, and getting an unhealthy horse that's hard to resell or use for anything is a problem, a vet check is a good idea.
Most people focus on checking a horse's legs and feet, but breathing, vision, heart should be checked, too. They are not always easy for non-vets to evaluate.
It's important, as someone said, to be sure the horse did not have/does not have founder (fever in the feet). Founder can permanently damage the feet and cause a lot of pain, some horses even need to be put to sleep as the pain can be severe. It's hard to identify founder unless you're familiar with the shape and appearance of a normal hoof. Foundered hooves tend to have rough, ringed, deformed hooves(but rings need not always be seen). The toes may turn and point upward or the hoof may look long, twisted. Founder doen't aways have visible signs on the hoof, but it usually does.
Another problem is lameness. Quite a few horses have arthritis at various points on their legs. This is a common cause of lameness. As he trots, a lame horse bobs his head, or hitches up one hip or hock higher than the other. He may stumble a lot (though there are other causes of stumbling). When it's not too bad, the horse might 'work out of it'. Some folks don't mind a little bit of arthritis in a family horse used for light riding.
