Basically, different breeds can produce offspring, whereas different species cannot. No matter how similar a guppy and a shark seem, they could never mate successfully, whereas a yorkie and st bernard could.
A species is usually defined by its ability to reproduce, therefore no matter how close genetically we are to gorillas (for example) one could not impregnate you, making it a seperate species.
A notable execption are horses and donkeys, which are sepreate species but still able to reproduce. Their offspring (mules or gennets depending on gender) are always infertile, thus preventing them from being a species in their own right.