From Jonathan Gardner's Korean Notebook
What is Romanization?
Romanization is different methods of writing the Korean words using the Roman letters---the letters that English and most of Western Europe uses.
There have been different Romanization schemes in the past. Some are better than others. The ultimate goal, of course, is to have the majority of people who use Latin letters read the Romanization and pronounce it correctly. This goal is, for various reasons, unachievable, mostly because the pronunciation of Korean is dramatically different from any Western European language.
My honest opinion is this. If you're going to live in Korea, learn Hangul. It's not hard and it will save you a lot of trouble and, at the same time, introduce you to the pronunciation system that Koreans use. Hangul is probably the easiest writing system in the world.
If you're just curious about Korean, learn Hangul.
If you don't care about Korean, then don't worry about it. You won't pronounce the words correctly anyway, so why bother?
The Revised Romanization is popular nowadays probably because it is the official Romanization method in use today in South Korea.
You'll note that it was instituted after my mission, so a lot of places I thought I knew how to spell are no longer correct.
There is also some complaint about the whole system because it doesn't work very well. Honestly, if you don't like the romanization, learn Hangul. It only takes an afternoon.
The older and (to me) more familiar romanization scheme is the McCune-Reischauer scheme. This is still in use in North Korea, but South Korea has pretty much abandoned it. You'll recognize it because it uses diacritics and other markings to distinguish between different sounds.
Again, if you don't like it, learn Hangul.
Wikipedia has a list of other systems.