They hold the shirt sleeve when putting on a coat, sweater or an outer garment.
They are for when you have on gloves. You loop your thumb through so no snow or wind gets down your sleeve while doing activities outdoors.
Simply 'Thumb Holes'.
Button holes
The holes may be button holes, air holes, or holes caused by damage to the shirt.
button holes arm holes head holes
sleve holes
because its a fashion
The proper way to check shirts for holes is to lay it out flat by a light. The light could be the sun, a lamp, or even a flashlight. Then you inspect the shirt using the light.
If you are trying to make the thumb hole then you just sew a basic button hole (but larger of course)
Big Thumb
Thumb on the back hole and the first three holes on the front.
Some dress shirts commonly have holes, which are pockets. The pockets are just a apart of the shirt for stylish accent.
It depends on the shirt. Cotton shirts are of course made from cotton fibers, synthetic shirts can be made from other things such as polyester and silk.