In woodwork, quartering is a form of veneering, in which a square section of wood cut against the grain to show up the grain, is cut into four very thin slices. Therefore the grain pattern is almost identical on each of these. They are then arranged in one big square, with one slice rotated by one turn, another by two, and another by three, so that the centre of the big square is a common corner of the four. This means the grain pattern radiates away from it and is symmetrically, almost kaleidoscopically, repeated.
This technique was introduced into English furniture in the William and Mary period (late seventeenth century) and remained popular into the 1720s.
In heraldry, quartering is the practice of displaying multiple coats of arms on one shield by dividing it in quarters and alternating the designs. If a wife brought her own estate into a marriage, the husband would want to display her (or rather her father's) arms with his own. His would be used in the first and fourth quarters, hers in the second and third quarters. The first quarter is the upper left (as we see it), though called dexter (right) because it's on the right of the person holding the shield.
One individual quarter, or the coat of arms on it, is also called a quarterng.
The oldest method of joining two coats of arms was dimidiation. This simply cut the two designs in half and stuck the left half of one onto the right half of the other, thereby possibly losing some important detail. This also created some peculiar objects like a lion dimidiated with the hulk of a ship. The next method was impalement, meaning the shield was divided in half vertically but both full designs were squeezed onto it, with consequent distortion. This method was replaced by quartering. In a quartered shield, the upper two quarters are fairly square, so there is little distortion, though the less important lower quarters do squash the designs, depending on the shape of the shield.
A quartering of two coats of arms is the simplest, for example the older kings of England bore 1 and 4 France, 2 and 3 England (since they also claimed the French throne). The modern British royal arms have 1 and 4 England, 2 Scotland, and 3 Ireland. But the number of shields simultaneously displayable is not limited to four. Firstly, you can just increase it to five, six or even more quarterings, and they're still called quarterings.
Secondly, one quartering is inherited from one ancestor, and that ancestor might themself have a quartered coat, so you get quarters within quarters. As the centuries went on some noble families got very complicated arrangements. In extremely rare cases the bluest of bluebloods might be able to claim seize-quartiers, 'sixteen quarters', meaning that each of their sixteen great-great-grandparents was entitled to bear arms; and even then, they wouldn't be allowed to show all sixteen on the one shield unless they were the inheritor (eldest son, usually) of each of the coats of arms.
And yes, it's one of the nasty bits in drawing and quartering.