No, a zebra is not a type of horse. Both are members of the genus Equus, but they are different species. Horses are Equus Caballus, Zebras actually have their own subspecies, Equus Hippotigris, and within that have three different species.
Like horses and donkeys (also different species within the genus equus) can be bred to make mules or hinneys, horses and zebras can be bred together to form a hybrid, usually called a "Zorse" or if crossed with a pony, a "Zony"-- these hybrids are sterile (cannot reproduce) because horses and zebras have a different number of chromosomes, so their hybrid offspring have an odd number.