How many types of roses
Welcome to the forum Stephanie, this will be a long answer. It begins at the Genus Rosa, which all roses fall under, then Species roses, which are the roses that occurred naturally with no human intervention. Then roses are broken down by Classes. Next would be OLD GARDEN: Alba, Ayrshire, Bourbon, Boursalt, Centifolia, China, Damask, Eglanteria, Gallica, Hybrid Bracteata, Hybrid China, Hybrid Foetida, Hybrid Multiflora, Hybrid Perpetual, Hybrid Sempervirens, Hybrid Setigera, Hybrid Spinossima, Moss, Noisette, Portland, Tea.
MODERN ROSES: English Style Shrubs, Floribunda, Grandiflora, Ground Cover, Hybrid Gigantea, Hybrid Moyesii, Hybrid Musk, Hybrid Rugosa, Hybrid Tea (includes climbing), Hybrid Wichurana, Kordesii, Landscape Shrub, Large-Flowered Climber, Miniature & Climbing, Miniflora & Climbing, Polyantha, Rambler.
(If you have insomnia you could do an internet search on each class, would be riveting at midnight.)
Probably more information than you need, however, if you have a more specific question, please let us know.
This information is available in the ARS Handbook for Selecting Roses. In this publication you can look up the names of many roses and their garden ratings.