Strictly, 'outboard' is the name given to ANY equipment which is external to the mixing desk itself. However, these days, it refers solely to the world of "effects and processors"; the 19" racks of equipment found to the side or (if you're well equipped) all around the FOH desk, used to shape and effect the sound.
An effect is added to an existing signal, like reverb and echo,
usually configured on an aux and return..
A processor processes the signal, changing it from the original, almost
always configured to the inserts of a particular channel.
Always get the entire mix sounding as close to how you want it to, before adding any effects (except, perhaps gates on drumkits). This means balance, levels and EQ should all be spot-on before you reach sideways. Generally, vocal compression will be first, as the vocal track will almost always have a significantly large dynamic range. Next will be vocal reverb; remember you might need a gate inserted on the vocal channel, but not always (and you can add it later).
Not strictly always outboard,
because all desks have a dedicated EQ on each channel. Theory is the same
regardless of if using a 3-band or a 31band/parametric.
Generally, apply cut to frequencies that are too present, rather than boosting frequencies that are 'lacking'.
Actually works by changing the phase as well as boosting the amplitude of the selected frequencies, this is why simply EQ'ing can 'bring out' the sound or set it back; the phase change emulates that that occurs in sound transmission over a distance.
When setting an FX unit that returns on another channel, an easy mistake is to set up the send level to low, and you increase the return level to compensate, you get lots of noise and it sucks. Instead, increase the send level (aux. out) to maximum possible without distortion (about '7' if you 0dB and PFL'd correctly).
After returning an effect into it's own desk channel, (like reverb), set the aux pre/post fade for that channel to pre fade, so that you can further process the signal, without HAVING to feed it into the mix.
Effects are virtually always configured on a post-fade aux, so effect only gets a signal when channel fader is up. The level of the aux send then configures the balance between 'straight'(dry) and 'effected'(wet) signal.
Also, effects are mostly configured so they only output 100% wet sound, despite the on-board ability to output an amount of the original (dry) signal as well. You will use the desk as a source of the dry signal. In the unlikely case of not having any spare channels OR aux returns, you can balance wet/dry using this onboard control, if you have to.
Ground loops can occur from rack framework, as well as the power supply and signal cabling.
Really common, but really dumb thing to do, is mount the PSU for the desk in the bottom of the FX rack. Not in itself a problem, but the PSU has a big fuck-off tordial transformer inside, so radiates lots of fluxing magnetic field. FX units are fragile at best, and don't 'process' in a balanced domain, so are extra-susceptible to interference. Very often causes people to think that good FX units are noisy because they perceive hum even when no input is connected.