Potato comes from a number of wild species found in various parts of South and Central America.
It seems that hybrids of these were domesticated around 5000 to 2000 BC, by people in highland areas, especially the high plateau around Lake Titicaca -- now Bolivia and Peru.
There are still hundreds of potato varieties in that area -- the ones we see elsewhere would be regarded by them as all more or less the same type.
There are several other domesticated plants related to the potato, including Capsicum peppers (red pepper, paprika, chilli etc), aubergine, tomato, garden huckleberry and tobacco.