How many $\alpha \in S_n$ are such that $\alpha^2 = 1$?

The way I was taught it, there are $$ \frac{n(n-1)}{2} $$ 2-cycles, $$ \frac{n(n-1)(n-2)(n-3)}{2^2 \cdot 2} $$ products of two disjoint 2-cycles, and in general $$ \frac{n(n-1) \cdot \dots \cdot (n-2k+2)(n - 2 k +1)}{2^k \cdot k!} $$ products of $k$ disjoint 2-cycles, provided $2 k \le n$.

These permutations are called involutions. The counting function for involutions on $n$ elements is documented at OEIS here. You'll be able to find explicit formulas, recurrence relations, asymptotics, and generating functions there, along with some references. OEIS (Online Encyclopedia of Integer Sequences) is a pretty nice resource in general for finding what's known about various integers, especially if you can compute the first several terms to search for.