Basically Koreans use Hanguel system only. Though the sound is written through Hangeul, lots of words are originally made up of Hanja. When your Korean gets improved a lot, it'll be very helpful to understand what words mean if you understand Hanja.

There is NO korean alphabet, because it does not derive from the greek and latin alphabets!! There are two writing systems, the 한글 (hangeul) and the 漢字 (한자, hanja). The former is nowadays the dominant one, but koreans still learn some 漢字. It is common to find 漢字 in newspapers, research articles etc. Sometimes it is used as an abbreviation, for example, instead of writing china and korea (중국 - 대한민국) (as in "the relationship between China and Korea") it would be written 中韓 in a newspaper. Another use of 漢字 would be to disambiguate homophones. For example, if I wrote 금화, it could mean, among other things, a gold coin (金貨) or a golden flower (金花). And of course, in the arts you find lots of examples of 漢字, for example in calligraphy (서예, 書藝). As you may know, half of the words in korean are loan words from chinese, so when you look at the 漢字 you can get an idea of their etymology. I really enjoy learning hanja because it has allowed me to learn lots of new words almost effortlessly. Some people argue that you don't need to know hanja in order to speak korean (even some native speakers think so), but in my opinion, it is like learning just one half of korean. It would be like learning english and not caring about Shakespeare, Chaucer or Beowulf.

There is only one Korean alphabet and it is written using Hangul through spellings. Hanja are Chinese Characters used to write whole words, From what I have seen from my own study and experience and from what I have heard from Koreans, Hanja is pretty much obsolete now in Korean, and everything is written in Hangul. I'm not Korean though lol.