Picking cotton now is all by machine. There is the harvester, and when it gets full it dumps its load into what I call a tamper( I don't know what it really is called) which tamps the cotton down real hard. When the 'tamper' is full it raises up and leaves a 'brick' of tamped cotton behind.
The farmers cover the cotton to keep it dry until a special truck comes by to carry it off to the cotton gin. At the cotton gin the cotton seeds are removed.
Cotton seeds have many uses- they are pressed for cotton seed oil, ground into meal for use as a soil amendment and of course some are saved for next years crop.