Horses cannot catch colds from people and there is no equine "common cold" equivalent.
Horses can, of course, catch the flu (equine influenza). They can also catch something called the equine herpes virus, which is the closest to a "common cold" - but it's much more serious. In fact, a horse diagnosed with "flu" might have either equine influenza or equine herpes - the clinical signs can be hard to determine and the treatment is the same.
The equine herpes virus comes in two strains. EHV-4 is the less serious of the two and tends to occur annually in foals (which have little immunity). EHV-1 causes miscarriages in pregnant mares and can affect the horse's central nervous system - which is commonly fatal. There are vaccines available and most adult horses are latently infected with (and thus immune to) both strains.
But there's no horse equivalent to the extremely mild...and extremely quick mutating.. "common cold."