We generally talk about our consoles having inputs and outputs. The mics, DIs, and other inputs come into the channel strips, and then we send mixes to Auxes, Subgroups, or the Main Mix, each of which is an output buss. And if we want to create a separate mix using those outputs, we use a Matrix.
When we get to the size of production that requires a separate console for the monitors, we also need a splitter. This takes the Inputs from stage and splits them to both the FOH and Monitor consoles. There are a few methods for doing this, but the one I prefer for live music is a transformer isolated splitter.
One of the things that's not so fun about theaters is that the FOH mix position is usually stuck underneath a balcony. This means that you can't really hear the whole PA (sometimes almost none of it), and the low/low mid frequencies are probably way louder than they are for the rest of the audience.
The biggest development in speaker systems for concerts in the last 20 years has been the line array. In pro touring, these are flown from truss or from a point in the ceiling above the stage, and they give precise control of where the sound is going. They also keep the sound away from the stage, which helps make the stage sound cleaner.
I was pretty excited when d&b, who makes some of the best speaker systems in the world for live sound, wanted to share this post about monitors. Obviously, it's pretty important for performers to hear themselves and this video shows how we do it, with wedges and IEMs. Music is interactive, and not hearing yourself or the other musicians can make being on stage pretty uncomfortable.