"Борзая" is an adjective that became noun. It is a short form of "борзая собака" expression.
Like bathroom - "ванная комната". "Ванная" is an adjective, that describes room. With time people began to skip "room" word and "ванная" started to mean "bathroom".
This is a special category noun-from-an-adjective in Russian. A word looks like an adjective and grammatically changes as an adjective (because it is an adjective in fact); but it is a noun and plays a role of a noun in the sentence. For example it doesn't change gender as an adjective; it is always feminine.

борзая is feminine, because it means a dog, that is feminine in Russian.
So if you are speaking about a type of dogs, then "Она - борзАя" is equal to "Она - борзАя собака".

Also there is an jargon adjective бОрзый; it can change gender. "Он - борзый. Она - борзая."
Means swift + impudent. Usually describes such people as chavs )