Polar alignment is pointing an equatorial mount's main axis at the celestial pole. It has a reputation for difficulty that comes almost entirely from people trying to do it to imaging accuracy when all they actually need is visual accuracy.
What accuracy you actually need
For visual observing, within a degree or so is fine, which you get by sighting along the polar axis at Polaris. For imaging, within a few arcminutes, which needs either a polar scope or a software routine. The two tasks feel like the same task and are not, and conflating them is why beginners find the process forbidding.
Using a polar scope
Most equatorial mounts have a small telescope built through the polar axis with a reticle showing where Polaris should sit relative to the true pole. A phone app gives the current angle, you set the mount's latitude scale, and you adjust altitude and azimuth until Polaris sits on the mark. Five minutes, once you have done it twice.
Software alignment
Modern routines take two or three images of the sky, plate solve them and tell you exactly how far off you are and which way to adjust. They are more accurate than a polar scope, they work when Polaris is not visible at all, and they are now standard on imaging setups. They need a camera and a computer, which visual observers do not carry.
The southern hemisphere problem
There is no bright pole star in the south, so polar alignment there has always been harder and has relied on drift methods or on the pattern of faint stars near the pole. Software plate solving removed that asymmetry entirely, which is one of the quieter improvements of the last decade of amateur astronomy.
Questions people ask about polar alignment
How do I polar align a telescope?
Set the mount's latitude to your own, point the polar axis north, and put Polaris on the right spot in the polar scope reticle using a phone app for its current position.
How accurate does polar alignment need to be?
Within a degree for visual observing. Within a few arcminutes for long exposure imaging, which usually means using software.
What is a polar scope?
A small sighting telescope through the mount's polar axis with a reticle showing where the pole star should sit relative to the true pole.