A headphone amp allows you to get the most out of your headphones. A lot of audio gear, portable players, and computers were not designed to drive higher quality headphones. They are fine with earbuds and consumer headphones, but once you start getting into better stuff plugging in directly may not be making the most of your better headphones.
Headphone amps are designed specifically to drive headphones. Generally speaking, the higher quality headphones tend to benefit, or sometimes even require a headphone amp to get the most out of them. If you are currently plugging your cans directly into a laptop, a headphone amp with built in DAC would allow a much better listening experience then an integrated sound card that was most likely never designed for critical music listening.
When trying to pick out an amp for your headphones, pay attention to the impedance rating of your cans and what the headphone amp supports.
Again, for your selected ATH-M20x a headphone amp is probably not a necessity, but it is good to know there is a possibility of further improving the sound from these cans. If you are interested in getting an amp for these cans, look for ones that are comparable value to your headphones. Not much sense getting a $500 amp for a $37 pair of headphones. Shop at a store that has good return policy - if you can't hear a difference, just return the amp.