DI stands for "direct injection" or "direct input." At its simplest, the DI port has two basic functions. First, it can serve as an impedance-matching device, converting a high-impedance instrument signal, such as a bass or a guitar, and converting it to a low-impedance signal, suitable for mixer input. Secondly, the DI port also converts an unbalanced signal to a balanced signal that can be run over long distances without signal loss.
DI ports come in both passive design -- those that do not require an external power source -- and active design -- which does require power, supplied by a battery or phantom power from an audio mixer. Passive DIs use a transformer exclusively to convert the signal, while active DI ports can use transformers, but more affordable designs often do not include them. This is not a drawback unless the strongest electrical isolation is required.
DI ports are commonly used with electric bass to obtain a clean and clear signal, free of speaker and amp issues. While a microphone on an amp is preferred for guitars, DI ports can send a dry guitar signal to the recording device, while sending a signal to the amp for conventional recording, acting as a signal splitter. Electronic keyboards may use DIs to limit the run of unbalanced cable, which may degrade the audio signal.
Connecting instruments live is done the same way and for the same reason as in the studio. The electrical isolation of a transformer design DI port becomes more important in a live music setting. The transformer creates a break in the signal path, preventing ground loops between the various amplifiers involved, reducing overall noise and preventing potentially harmful voltages being fed back through guitar strings and microphones. Using the ground lift switch on a DI port may also kill an annoying hum that cannot otherwise be removed.