Hard subtitles are permanently connected to a video. You never have to worry about not being able to see them but that also means you can't turn them off or change how they look. Soft subtitles are recommended.
Soft subtitles are one or more files alongside video files that allow you to view them or not. This can be a boon to users who don't have the correct software installed but is easily resolved. There are two kinds, text and binary subtitles. Text subtitles (usually ending in .SRT) allow you to edit them easily while binary subtitles (usually two files ending in .SUB and .IDX) can include multiple languages in one file but cannot be edited.
There are a lot of websites where you can view or download movies and dramas with subtitles. For a list of sites that have Korean movies and dramas, see Studying resources.