Where do new stems and shoots come from after pruning?

If I prune a rose and cut to a bud, is it from that very bud that the next shoot will emerge? If it is only a vegetative bud, only foliage will appear, and a flowering shoot will set forth only flowers and fruit, so how does a plant go from having its stems cut short to having them re-grow/

If dieback occurs at the place you made the cut, the stem cannot re-grow its self. If it grows out of a fresh bud, how come all gardens' plants are not slanting to one side, as each year the new growths are at an angle?

Or does the stem re-grow?

Thanks in advance.