The Korean language did not grow in a vacuum. It was influenced by its neighbours: Japan and China. Hanja is China’s biggest contribution.
Hanja and Hanja-eo
When you have spent some time learning Korean, you might have noticed a peculiar thing. A lot of syllables tend to come back and those syllables return in words that are connected in meaning somehow. There is a very simple explanation for that and that explanation is to be found not in Korea, but in China.
Long ago Hangeul (한글) did not exist, but that does not mean Koreans did not write things down. However, when writing things down the ancient Koreans used Chinese characters which were called Hanja (한자); though it is a far complex situation than just that. When those characters are used to represent the Korean language, they are called Hanja-oe (한자어). The use of these characters would last well after the inception of Hangeul. In fact, it is quite the recent evolution that Hangul is used dominantly in Korean writing. Nonetheless Hanja is still used today and its use is even make a little revival.
The influence of Chinese, though, did not stop with its characters. China also donated a lot of its vocabulary to Korea. One of its donations you probably know very well are the Sino-Korean numbers. Yet, there are countless Sino-Korean words out there and you will have undoubtedly learned many of them without knowing it.
Difficult, but useful
One of the reasons why Hangeul was developed was the difficulty level that came with the use of Hanja. Not only were there thousands of individual characters, but Chinese characters were meant for the Chinese language, not the Korean language. It was very difficult at times to represent the Koreans language with these foreign characters. This among other reasons inspired King Sejong the Great to develop Hangeul and it really succeeded in simplifying writing in Korean.
However, Hanja is still very useful in Korean. There are two uses for Hanja.
Understanding unknown words
The first reason can be a great tool for those wanting to increase their passive knowledge of Korean. By understanding the Chinese characters to represent a word, you can guess the meaning of others that use the same characters.
Let us take 학샣 (student), a word most of the Korean learns will know. If we were to write it in hanja, it would become 學生. Each character corresponds to a part of the word. The character 學, means learnings, knowledge; school. Now what if I were to tell you that 학교 is 學校 in hanja. Notice something. Indeed, the character 學 is represent the same syllable (학). This is no coincidence. You will find a lot of words that are related to learning, knowledge or schools have this character (and syllable) somewhere. This can become pretty useful.
However, I think Jeremy from Motivate Korean does a better job explaining it than me. He also shares a ling to a very nifty hanja dictionary.
If you have have a few Korean words under your belt, you might have noticed something, something that can be problematic. It is not uncommon for Korean words to have a lot of meanings. Sometimes those meanings can be vastly different. That is quite problematic and Koreans know this. However, there is a way to make without a doubt clear which meaning you mean. You guessed it, by using Hanja.
Koreans do not use Hanja like they did in the days of the ancient Korean kingdoms, but now and then they still write a thing or two in Chinese characters. They do this to avoid confusion when it is better to avoid confusion. In newspapers you will sometimes see Hanja as well, but here the use is not so much to avoid confusion, but to help Koreans understand some uncommon words. As mentioned above, Chinese characters can make the meaning quite clear even if you do not know the exact meaning.
Start Learning Hanja
As you might have noticed Hanja can be very useful when you know it, but I would not advise a beginner to start off with these Chinese characters immediately. There is a reason why the Great King created Hangeul in the first place. There is a mountains of characters and they might very well overwhelm you. Therefore, if you are a beginners, I would advise you not to start actively learning these nifty characters. Try to keep it in back of your mind, though.
After you have gotten familiar with Korean and feel like you are ready for that little extra, then I would say: Go ahead, have fun! As mentioned, Hanja can be a real treat to vastly increase your passive vocabulary knowledge. It might even speed up your reading comprehension since you will use a dictionary less often as you will be able to guess the meaning by merely looking at the word. Now that is neat, no?
If you are learning Hanja, let me know in the comments when you started learning the characters and if they have been of any use to you since you started. I am quite curious to hear about your experience.