"Current" means "now". But "now" can't be used as an adjective. So we use the word "current" instead. In the example above, the "current President" is the person who is President of South Korea right now. Here are some other examples:

He had two kids with his ex-wife, but he doesn't have any with his current wife.

Their current computer system is really old and hard to use, so they're looking to replace it.