Signal chain is the path that the signal takes from your instrument to speaker (where it's heard by an audience) or sound board (where it's mixed with the signals from other instruments before being amplified to an audience). Your signal chain can become long and complex as you add in multiple effects and preamps but the the basis of any signal chain for a musician using a PA is:
1/4" In from instrument > DI box > XLR Out to soundboard
Instrument cable, guitar cable and 1/4" (quarter inch) cable all refer to the same kind of cable and plug. 1/4" can be either mono or stereo, but for plugging an instrument into another step in your signal chain you will need a mono cable.
Mono cable (1/4" TS)
A mono cable connector has a tip and a sleeve (TS), which are each soldered to their own wire in the cable. Mono cables send or receive just one unbalanced signal.
Stereo cable (1/4" TRS)
A stereo cable connector has a tip, a ring and a sleeve (TRS), adding a third wire to the cable. This allows you to pass two signals out of one source in stereo. It can also be used to send just one balanced signal or to send and receive signal in an effects loop.
Impedance, abbreviated "Z", has to do with the amount of resistance in an electrical circuit, but for practical musical purposes you just need to know that electric bowed strings instruments have a very high impedance, or Hi-Z, signal.
Sound boards don't accept Hi-Z signals, so players of Hi-Z instruments such as the violin need to add a stage in their signal chain that will transform their Hi-Z signal into a Lo-Z signal that can plug into the sound board.
An XLR cable, also known as a mic cable, sends a balanced signal. It has a male end with three small prongs and a female end with three corresponding holes. These are connected to three wires inside the cable--a positive, a negative and a ground. An balanced XLR cable is what you will use to connect to the sound board from a DI box...
Direct inject is commonly abbreviated and always referred to in live sound as simply "DI". A DI box is a piece of equipment that the contains circuitry required to transform your Hi-Z input into a Lo-Z output that can plug into a sound board.
A passive DI box is simply a box with some transformers inside and requires no power source. It is simply converting signal and sending it along the chain. These units are typically inexpensive.
An active DI box performs the same Hi-Z to Lo-Z transformation but includes other features that require a power source. Active DIs often double as a preamp stage and have a gain boost. They may also have EQ (equalization) and feedback control features. Power can typically come from either a 9-volt battery in the unit or from phantom power.
Phantom power is power supplied to a DI box or microphone from the sound board, traveling back through the neutral line in the XLR cable. Troubleshooting tip: an XLR cable that may work perfectly well with a passive DI box could then fail to work with an active DI running on phantom power. The reason for this could be a break in the neutral wire that supplies the phantom power.