A modified underground STEM that grows laterally from the rootstock/mother plant/crown (usually no more than a few inches below the surface) is considered a rhizome. All along the surface of the rhizome there will be rings of buds spaced about 4-6 inches apart. The buds are sometimes called 'eyes' just like the ones you see on a potato. The rhizomes that folks buy are mostly just a chunk of that underground stem which have been cut into sections about 6 inches long and contain at least two rings of buds. When you plant the rhizome, it's only source of energy for growth are the carbohydrates it has stored up inside itself and uses this energy to form roots which can then obtain nutrition from the surrounding soil. At the same time, the buds will turn into little shoots which will grow into the vines that will climb and produce your crop. The leaves on these vines will also be able to take sunlight energy and turn it into a form that the plant can use. All this happens in the first year that the rhizome is planted.
Once the vines die back in the fall, the underground portion will continue to live, and being that it now has a developed root system, you can pretty much call it a crown. Over the course of the past growing season, more buds will have formed on the crown which will sprout and turn into vines again next year. The huge difference in year two is that now it doesn't have to spend time and energy to form roots and can concentrate on growing vines and making hops for us to brew with.
Depending on growing conditions and varieties, some may have enough stored energy to begin to form their own rhizomes in year two but in the 3rd year you can usually count on there being some rhizomes formed. No matter how old the plant is, as long as you dig and remove any part of it that has buds attached, you can essentially remove it without worrying that it will come back (unless you missed a bud or two). Again, it's very easy to remove the whole plant after one season without leaving anything behind that will regrow. Hope this helps.