So what does a patch bay do? It gives you a place to bring all of your signal sources and destinations together, so that you can connect things in any arbitrary order using short patch cords. In that respect, it's no different from a modular synthesizer. However, a well laid out patch bay can also make your life easier by having pre-made "default" connections that correspond to the way you normally connect things in the studio, but can be easily overriden when desired. And, by making it more convenient to change connections, it can enable the use of things that might otherwise be too much trouble to cable up -- like auxiliary outputs from synths.
Your standard 19" rack-mount patch bay consists of two rows of 24 1/4" jacks each on the front, with a like amount on the rear. (Some pro patch bays use a different connector called a "TT" connector, which looks like a 1/4" connector but is smaller; it's about 1/6" in diameter. You'll probably want to avoid those for a home studio, because you obviously have to buy different cables for them.) Each jack in the rear corresponds to a jack in the front; there's a direct connection. The rear is where you make your (relatively) permanent connections: you bring all of your signal inputs and outputs to jacks in the rear. Each connection in the rear causes the connected input or output to appear at the corresponding jack in the front.
(Some patch bays designed for large-studio use have other connection means in the rear, such as large multi-pin connectors, punch-down terminals, or lugs for soldered connections. These are used by pro studios because they are more reliable means of connection. They do, however, make it much more difficult to change things around. Home studio users should probably stick to the patch bays that have 1/4" jacks in the rear.)
The patch bay is a completely passive device, requiring no power, and so any jack can be used as either an input or an output. However, by convention, outputs are connected to the top row, and inputs to the bottom row. And the patch bay is designed with this in mind: the process of setting up those "default" connections relies on it. In recording engineering terminology, the process of setting up default connections in a patch bay is called normalling. When a "behind the panel" connection exists between a top row jack and the jack below it, the two are said to be normalled together. Any such pair of jacks can be, with no patch cords connected, in one of three states: full normalled, half normalled, or non-normalled.
As you might expect, non-normalled means that no default connection exists. The difference between full normalled and half-normalled is this: In a full normalled setup, whenever a patch cord is inserted into either the output (top jack) or the input (bottom jack), the normalled connection is broken. Inserting a cord into the output jack routes the output to the cord and leaves the input jack connected to nothing, unless a patch cord is inserted there also. Similarly, inserting a patch cord into the bottom jack connects the signal from the cord to the input, and the ouput is left connected to nothing until and unless a patch cord is inserted there also. In a half-normalled setup, the bottom jack behaves the same, but the top jack behaves differently: inserting a patch cord there does not break the normalled connection. Assuming that the bottom jack has no patch cord inserted, then inserting a patch cord in the top jack effectively creates a Y-connection: the output signal is now routed to both the normalled input and the inserted patch cord.
Here's a tip: Less expensive patch panels usually don't support full normalled connections; the choices are half normalled and non-normalled. That's okay, because as it turns out, full normalled connections are not all that useful. When you patch into an output jack, creating a Y-connection is often what you want. And if you don't, it's no big deal to insert an unconnected cord or a shorting plug into the corresponding input to interrupt the normalled connection. Full normalling just doens't buy you much over half normalling, at least not for electronic music and home-studio purposes.
Nearly all patch panels let you choose the normalling on a per-jack-pair basis. The mechanism for doing this varies. Below you see a photo of my two patch panels:
And the rear:
Both of these are inexpensive models, but the top one is somewhat less expensive. As you can see from the rear photo, it's an "open frame" design, with the case not fully enclosed. There's a reason for this: Each set of four jacks (two front, two rear) is on its own circuit card. The card is held in by a retaining nut that clamps it against the retaining strip in the rear. To change a card from half-normalled to non-normalled or vice versa, you undo the nut, pull the card out, flip it around, and put it back in. This makes it very easy to change. The jacks and retaining nuts are all plastic. It's not the most rugged patch bay in the world, but for home studio use it's fine. It came with one spare card, so I have a replacement if one goes bad, although there isn't much that can go wrong with it.
The bottom one is a little bit higher-priced. It's more rugged, with all metal construction. Like the top bay, jack pair normalling is changed by flipping the card. Problem: To get at the cards, you have to remove the front panel. Since the front panel is what holds it in the rack, what this means in effect is a massive disassembly job whenever you want to change the normalling. To save a little time (and avoid having to undo and redo all of the rear connections), I've developed the trick of leaving it mounted and detaching the body of the patch bay from the front panel by undoing the hex screws that you see in the front. Then, I can pull the body a few inches away and get at the cards. See below:
However, getting it reattached, lining up the screw holes and all 48 front-panel jacks, is quite a balancing act. So if you need to change the normalling on a pair, it's a big time-consuming production. Because of that, I tend to use the bottom bay for more permanent things, like mixer inputs and effects sends/returns, and I use the top panel for things that get changed more often.