How do they check really large primes? Currently, the largest prime know is a mersenne, $2^{82,589,933} − 1$. That’s an $82,589,933$-bit number if I am correct. Considering that RSA codes of as low as 1024 bits can be considered safe, how was this number factored to check if it is prime? I can kind of answer that question myself, I am aware of the existence of a special, much faster, prime check for Mersenne primes. But, given a non special number of similar size, would we even be able to check if it was prime? How long would it take? How fast are the fastest prime checking algorithms for non-special form numbers?

Mersenne numbers are special as others have pointed out (the proof of primality is done with the Lucas-Lehmer Test). Fermat numbers are also special and can be proven prime with Pepin's Test. There are also well known primality tests to use for numbers of the form $k2^n±1$ when $k<2^n$ (Proth's Theorem for the plus side, and a related test for the minus side using Lucas Sequences). I recommend reading from here if you are interested in these tests.
RSA-primes on the other hand don't use deterministic primality tests like the ones above. Instead (in most cases), one uses probabilistic tests (they work well in practice, but cannot prove that a number is actually prime). Such tests include Fermat test, Miller-Rabin, Euler-Jacobi, BPSW, Frobenius, etc. 
If provable primes are desired, it is possible to prove RSA-primes 'prime', but will come at a timely cost (300-700 digits roughly speaking). The quickest of methods that are used include APR-CL and ECPP. Still, these become impractical (timewise) once the size of the input is about 10k or 50k digits or so, thus there is no current (practical) method to prove such large numbers primes and we must either resort to probable primes (any number), or provable primes which rely on (some)  factorization of usually either $n-1$, $n+1$ or both.
