Vi liker begge to både å svømme og løpe = We both like to both swim and run. Begge likte både bilen og sykkelen = Both enjoyed both the car and the bike.
It's "både - og" meaning not only, but also. You can't ever have just "både" as it's a comparative. While "begge (to)" is sort of a pronoun (actually a determinative) meaning two.
They both (sic) stem from the old Norse báðir, "både" being the neutral (it says "nøytrum" in my preferred online dictionary but I fail to find a term for it in my Oxford advanced learner's dictionary) and "begge" a genitive.