Is the number $-1$ prime? From my understanding it's not prime because it's not greater than $0$. So my followup question is why did mathematicians exclude $-1$?
The definition of prime is having only two factors. 
$-1 \cdot 1 = -1$

In general, a definition that describes relationships between numbers without describing the domain in which such relationships are valid is incomplete.
In other words, when you say that the "definition of prime is having only two factors," you are leaving out a crucial element: for which set(s) of numbers is this relationship between primes and factors valid?
The typical definition of "prime numbers" describes a relationship between natural numbers; since $-1$ is not a natural number, any definition that only applies to natural numbers has no bearing on whether $-1$ is prime.
Thus, in order to make the question "is $-1$ prime" meaningful, you must find an extension (or generalization) of the definition of "prime" to a domain that includes $-1$.
As pointed out in @Hugh's answer, perhaps the most natural extension is that of prime elements. In this extension, $-1$ is considered a unit and is therefore by definition not prime. Another extension is 
the Gaussian Primes, in which $-1 = -1 + 0i$ is non-prime because the coefficient of $i$ is $0$ and $|-1| = 1$ is not prime (according to the standard definition on the natural numbers). @BillDubuque, meanwhile, gives an example of a context in which it might be reasonable to consider $-1$ to be prime.
Now, we can perhaps make further generalizations about all possible extensions of the definition of primes within the natural numbers. Some of the other answers provide some good intuition about this; for instance, it is observed that the units are typically not considered prime, because this simplifies the definition of prime factorization. Thus, we could answer that $-1$ is not considered prime in any domain in which $-1$ is a unit (though of course this requires us to define "unit" for each such domain).
