I'm wondering what types of screws I should use for woodworking and what types I shouldn't be using as well.
For example, Are deck screws OK? What about a square head vs. Phillips? Does the type of metal/construction of the screw matter?
Just to get it out of the way, this is the conventional/traditional wood screw:
In the past it would always have been flat-head (now also called slot-head), and as a result when making a piece in a traditional style many woodworkers will insist on using flat-head screws so they look the part.
Are there specific types of screws I should use for woodworking?
In short, yes. But which type to pick is often on a project-by-project basis. While there are screws that were not originally intended for use in wood that are OK to use for specific purposes (when used appropriately, e.g. machine screws in jigs or tools), there are most definitely a few screws you shouldn't use in woodworking, and drywall screws top that list. Some people value them because they're sharp and drive well in the wood they're using, or for their black colour. But they are too brittle for woodworking in general and should never be used ideally.
Are deck screws OK?
For decks, sure.
But you need more detail than that :-) How they are intended to be used in that application is a big factor in how they should be used. Decking is, like most exterior woodworking, sort of an extension of rough carpentry. In any project in a similar vein they can be a good choice. Although they can of course be used indoors they're not the fastener of choice for interior use and particularly for furniture pieces other screws should be preferred (many of which would be cheaper anyway).
What about a square head vs. Phillips?
Yes and yes. Point to remember, these are not screw types, these are just head styles. A great many different types of screws can have either kind of head.
Square-drive screws — which I feel we should call Robertson screws to give due credit to its inventor, even if we're not Canadian — are a particularly good screw because the square shoulders are very strong and resistant to stripping. Also on a practical level the tight fit of the driver to the screw can be good enough that the screw can hold securely on an unmagnetised bit, aiding single-handed driving. This is a particularly good head style I think and it's a shame it's not more common; possibly the only reason they're not more widely used worldwide is that when invented manufacturing wasn't capable of producing the screw strongly enough in a cost-effective manner.
Phillips screws (sometimes referred to generically as cross-head) have a long track record and can work fine but are now a bit outmoded, with the PoziDriv (Pozi for short) screw being an improved version. One general principle with screws that any good guide will emphasise, that the driver should be carefully size-matched to the screw, is particularly important with Phillips screws. While you can use both a slightly undersize or oversize screwdriver with a Phillips-head screw (they'll both seem to fit well enough) there is a great potential for the shoulders to round over when the driver 'cams out' and if you do this just a few times the screw can strip completely. In a somewhat similar vein PoziDriv screwdrivers should never be used on Phillips-head screws.
Does the type of metal/construction of the screw matter?
Yes, very much so but this is too broad a question to go into much detail so I'll just randomly pick a few salient points.
Brass crews are considered a must for use with brass hinges and other brass fittings. Where steel screw are used, they stick out like a sore thumb visually and just give a cheaper appearance.
Brass screws must always go into pre-drilled holes, even in softwoods, or there is a great risk they'll snap when being driven in. In hardwoods in particular it is also good practice to drive in an exactly matching steel screw before driving the brass screw, so that the stronger steel screw can thread the hole and reduce the strain on the brass. Manufacturers of good brass hardware like Brusso will supply a steel screw in every pack of brass screws for exactly this purpose.
Even with this, some people use an extra security measure and lubricate the brass screw, using wax (just a candle will do), soap (any hard bar) or an oil or fat (e.g. lard or tallow, both used historically). Despite the steel screw being stronger it can be a good idea to lubricate it as well, particularly since it has to be driven in and removed multiple times.
Note: these days with the increasing cost of brass "brass screws" may not be solid brass but brass-plated steel. So while there's the potential for the brass plating on the head to be scratched off, they are stronger.
For exterior use, all screws and bolts should have a rust-resistant coating of one kind or another, or be made from stainless steel.
Conventional screw use
The traditional use of a screw was to fasten a board to another board, often to its edge, so one board is considered the thin board and the second the thick board (even when they are in fact the same thickness of material). And in this application there are some rules of thumb. The length of the screw should be approximately three times the thickness of the thin board being fixed. Older guides (early-20th century and before) often state that the pilot hole should be deeper than needed so that the screw doesn't 'bottom out', but modern guides do not agree on this point and it's rarely done today, however, in particularly hard woods this can still be a good idea.
In cross-section like this notice that the screw actually needs to go into two sizes of hole, the pilot hole we're all familiar with but a wider clearance hole in the first board. This drawing makes this clearer:
Even where some of the threading on the screw remains in the thin board the idea is that it is not held there, which can lead to a problem called bridging where no matter how hard you drive the screw the gap between the boards won't fully close.
Screws and end grain
Screws do not generally hold well in end grain. Where they must be driven into end grain you can 'toenail' or 'dovetail screw' them, that is, drive them in at a slight angle so that they partially bite into long grain. Another option is to glue in a dowel (preferably hardwood) in the path of the screw:
Some very coarsely-threaded screws, e.g. lag screws, are considered to have so much hold that this isn't an issue but they tend to be used where a very firm and reliable hold is needed and it's worth going the extra mile and adding the dowel. Or alternatively switching to a bolt and barrel nut, as show in this previous Answer or a conventional nut and bolt as shown in the first illustration here.
This one might be a little broad as it can greatly depend on the project.
In short no
There are hundreds of screw types, so sure, there are a bunch that would not work like drywall screws, sheet metal, roofing and pretty much all machine screws are out of the question as they will not bite properly or hold wood to wood1. Then you have to look at the part of the screw like the different head and types. Some are multi-purpose and other are meant to excel in certain materials and applications.
Are deck screws OK?
I use deck screws all the time as they just happen to be available for me and were designed to work outdoors (and with pressure treated wood which I use only for shelving and storage) which is where most of my projects end up. So for me it is an all around screw.
Wood screw, in general, are perfectly fine. That is not even a really specific category as there are different metals composites. That is true of all near all screws.
Does the type of metal/construction of the screw matter?
Yes, but it can change from project to project. Different metal compositions have different properties. Brass tend to have an allure for appearance and can cause less friction. Steel is valued for strength but is not as corrosion resistant compared to others. Stainless steel is a corrosion resistant (There are many grades there of as well). There are many other metals besides those with their owns strengths and fall backs.
While is possible to have the wrong screw for the job you need to overthink this. Features that are ideal and can be helpful for a wide range of projects would be self-drilling / tapping, countersink heads and Robertson or square head as they will be harder to strip of the common sockets (Personally I try to only use square head for that reason.). You can also get the combination Phillips and square head as well.
I see no issue with deck screws or basic wood screws for simple projects. Larger projects might need a little more consideration.
1 - I did make a simple planter box with drywall screws just to finish the container. It is still holding today just fine but I could have used better screws.
For general use screws, you can't beat the SPAX brand fasteners. They have a combination phillips/square drive head, drive well, and are quite strong.
For "fine" projects, I use brass screws for attaching hinges and catches, but none for joinery.
Generally, don't waste your time on the standard wood screws unless you absolutely have no choice.