What is the technical name for a lens which has the same maximum aperture throughout the entire zoom range?
This is a constant maximum aperture zoom lens. The "maximum" is often omitted — "constant-aperture zoom" — but is useful to include the word maximum to resolve a possible ambiguity with a fixed aperture lens, which is a much more rare design where there is no ability to stop down the lens from its one and only aperture setting.
Fixed aperture is only common in catadioptric lenses, which use mirrors, and I'm not aware of any that zoom, so in some sense the distinction is pedantic; everyone will know what you mean if you say just "constant-aperture". You will find people calling constant-maximum aperture zooms "fixed aperture", and if they're not talking about mirror lenses you can usually assume that they don't actually mean it.
constant maximum aperture zoom lens - Lets say the lens is a 2.8f constant aperture and focal length is variable from 35mm - 100mm. It simply means that at 30mm through 100mm the 2.8f is available to use. Many variable lens say 3.5 - 5.6f meaning 3.5f at the 35mm focal and decreases towards 5.6 as the lens is zoomed out towards 100mm. So at 100mm fully zoomed out the largest aperture available to use is 5.6 . All this means is you need more light at the longer end of the lens. So a 3.5 constant can shoot in lower light than a 3.5-5.6 lens can.