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Hard (also known as hardsubs or open subtitles). The subtitle text is irreversibly merged in original video frames, thus this format of subtitles doesn't require any special equipment or software at all. Thus, very complex transition effects and animation can be implemented, such as karaoke song lyrics following, various colors, fonts, sizes, etc. However, these subtitles can't be turned off unless the original video is also included in the distribution, because they are just a part of original frame, and it's impossible to do several variants of subtitling, for example, in multiple languages.
Prerendered subtitles are separate video frames that are overlaid on the original video stream while playing. Prerendered subtitles are used on DVD (though they are contained in the same file as video stream). Obviously, player is required to support such subtitles to display them, and it is possible to turn them off or have multiple languages subtitles and switch among them. On the other hand, subtitles are usually encoded as images with minimal bitrate and number of colors, thus they usually lack anti-aliasing font rasterization. Also, it is hard to change such subtitles, but special OCR software, such as SubRip exists to convert such subtitles to "soft" ones.
Soft (also known as softsubs or closed subtitles) are separate instructions, usually a specially marked up text with time stamps to be displayed during playback. It requires player support and, moreover, there are multiple incompatible (but usually reciprocally convertible) subtitle file formats. It's relatively easy to create and change such subtitles, and thus it's frequently used for fansubs. Text rendering quality can vary depending on player, but, generally, it's higher than prerendered subtitles. Also, some formats introduce text encoding troubles for end-user, especially if very different languages are used simultaneously (for example, Latin and Asian scripts).

Source(s):
Wikipedia