Tapping is probably one of the most misunderstood things I can think of on the home shop level. It is one of those tasks that is so easy to do if you're doing it correctly, but goes very wrong very fast if any one of the parameters is wrong.
The most common things I see wrong when tapping holes are, in no particular order:
Wrong size tap drill hole
Wrong tap style/material
Wrong feeds and speeds
Crooked lead in
Lack of proper lubrication/cutting oil
The biggest offender I think is cheap carbon steel hand taps. They are just garbage, almost always. Especially for power tapping. The metallurgy is sketchy at best, the design is not meant for repeated or continuous use and they tend to be fragile.
A hand tap is designed to neither push or pull the chips, but to curl them into the flute. This is why you have to stop, reverse the tap to break the chip, and then keep going. This is also why you can't power feed them. There is nowhere for the chips to go, and the get jammed up in the flutes and the tap breaks.
Taps designed to be power fed are usually either spiral point or spiral flute. The spiral point is for through holes, and will push the taps out the bottom of the hole. They tend to be durable, with only two or 3 flutes and are almost always HSS. Spiral flute taps pull the chips out the top of the hole and work best for blind holes. The are a bit more fragile due to their design, but are ideal if you have to thread into a shoulder or a shallow blind hole.
The other common production tap is a roll form tap. This tap doesn't make chips, rather it squishes the metal around it into the shape of the thread you want. This is especially effective in softer metals like aluminum, and it actually makes a really strong thread because instead of shearing the metals grain structure, you're simply molding it around the tap. These taps aren't really useable by hand, they like rigidity and power that the machine tool provides.
We tap thousands of holes in my shop, and there are probably as many ways to do it as there are holes.
If we have one or two holes to tap, we'll just chuck up a tap in the bridgeport spindle and power tap. The operator has his hand on the drum switch and reverses the spindle at the right moment.
If we have a few dozen or more holes to tap, we'll put the tapping head in the drill press. We have a Procunier tapping head. More on that here.
https://www.garagejournal.com/forum/threads/id-tap-that-new-procunier-tapping-head.438141/
If the hole is part of another machining operation, we will rigid tap in the CNC mill. Rigid tapping is where the CNC control synchronizes the feed of the spindle with the RPM of the tap, such that the spindle advances at the correct rate for the lead of the thread. It takes seconds to tap a hole in the CNC, as it is running a high horsepower spindle, has rigid and correct workholding, everything is aligned properly, and there is usually flood coolant or some other applied oil.
We only hand tap if there is no other choice. I will usually run a spiral point tap in a cordless drill if we can't hold the part in a machine. I don't even own hand taps, we use spiral point for most everything.
Union Butterfield, OSG and Greenfield make my favorite taps.