Equine: This term refers to anything related to horses. It's like saying 'automotive' when talking about things related to cars.
Domestication: This is the process by which humans have adapted wild animals (like the ancestral wild horse) for their own use. Imagine if you could take a wild tiger and over generations make it as friendly and useful as a house cat - that's what our ancestors did with horses!
Breed: A breed is a specific group of domestic animals having homogeneous appearance (phenotype), homogeneous behavior, and/or other characteristics that distinguish it from other organisms of the same species. Like how there are different models of cars (SUVs, sedans, sports cars), there are different breeds of horses (Arabians, Thoroughbreds, Clydesdales).