Threading is a wonderful thing. Without it
nuts, screws and bolts would not work. Without threading there would be no thumbscrews, no head bolts, in fact almost nothing in modern
technology could exist.
Of course for there to be threads, one must first create a tool for making said threads. That is where a tap and die set comes in handy. With one you can thread, or repair threads on almost anything. A tap is used to thread a hole, like in a nut or another object you must fasten. Taps thread female fastenings. A die threads male items like screws and studs. Each is is wound through (in the case of taps) or around (dies) the object to be threaded, which physically cuts threads into the object. Often a lubricant such as cutting oil is required. Tap sets often come with a properly sized set of drill bits to create the hole to be tapped.
Generally a different tap is required for each size and thread count. For example, one of the most common screws used in electrical installation is the 1/4-20 screw. That stands for a screw of 1/4" in diameter and 20 threads to the inch. Assume that you have made a part that you want held by a 1/4-20 screw. First you drill a 1/4' hole, then insert the tap for that size hole and turn under pressure until the opening is threaded. Dies are sometimes more flexible. The conduit threaders at work are often adjustable, with the same NPT threads serving many different sizes of conduit. As a tap or die is operated, long metal shavings will be produced, and they must be cleared occasionally. The process produces a lot of heat.
Full sets are rare in industrial use where the same tool is often used over and over. Mechanics will want a full set, primarily for repairs. Careless installation of a screw or bolt will often lead to cross-threading, where the screw or bolt is started at an angle. The threads of the nut and bolt cross each other somewhat, Most of the time cross-threaded objects won't start. But sometimes they do start, and a careless or impatient worker may force them to thread together. Doing that damages the threads on both nut and bolt, necessitating repair or replacement. A tap and die set makes repairs possible in many cases.
Anyone with significant experience in auto mechanics can tell you that the number and types of potential threadings is almost limitless. Engineers specify threading for their own dark purposes. Nevertheless, many threadings are quite common, like the aforementioned 1/4-20. A small tap set with drill bits may go for as little as US$20 on sale. A medium sized set of taps and dies might cost US$150. It is possible to spend thousands. My recommendation for a mechanic is that when you need a tap or die, to go ahead and buy a medium sized set. Such a set will offer most sizes through 1/2' (13mm) at two common threadings. Sooner or later you'll need the other pieces and if you have a set you won't have to stop and run to the hardware store. Then buy the real oddball pieces individually as needed.
Your set will also prove useful in identifying and cleaning threads. Taps and dies are marked. The right sized tool should thread easilly onto an unknown bolt or nut. That process often cleans the threads of foreign objects like paint which impede threading. If your set doesn't work I'd take the piece in question to the nearest hardware store and keep trying stuff until you find a good fit.
A tap and die set is not something you will use every day, like a socket set. But when you need one they are really handy. They are a wise purchase, but only after you have a fairly complete set of good hand tools. Buy one the first time you need to thread something and then you'll have it for next time. No well-equipped mechanic would go without one.
rootbeer277 reminded me that small taps break quite easily, particularly when used with an electric drill. The weight of the drill is often what breaks the tap, even if the clutch is set properly. It is wise to have spares of sizes you use all the time, and I often prefer a hand tool than a drill to protect my taps.