208V is most always a 3 phase system. 208V can be measured between phases A&B, B&C, and A&C. This is usually made when the transformer's (3) secondary coils (which each generate 120V across them) are connected in a "Y" (wye) fashion, the common center point of the Y (wye) is the "neutral" (which is usually grounded to earth; it doesn't have to be but that's another subject).
So now you have 3 phases, with each phase 120V to neutral (or ground), but with 208V between each phase.
Why do (3) 120V transformer coils connected this way result in 208V between each phase? That's a bit more complicated.
Note that the relationship of 208 to 120 is 1.73 (which is the square root of 3)
The voltage sine wave being induced on each coil is 120 degrees out of phase from the voltage sine wave on the other 2 coils. (+120, 0, -120)
(They're 120 degrees separated because back at the generator, 360 degrees of rotation was divided into developing "3 phases", 120 degrees apart from each other (I digress, don't be distracted)
There's a lot of mathematical ways to prove that vectors separated by 120 degrees of rotation have a "square root of 3" (1.73) relationship. (120/208, 277/480V, 347/600V)
I think it's easiest to see how like this:
Look at the letter "Y", where each "leg" of the "Y" is rotated 120 degrees from the other. Imagine the length of each of the 3 "legs" are 120 units (volts, or inches, miles, whatever). Now measure the straight line distance between the end of one "leg" to the end of another. This distance is 208. Thus you can "see" how (3) 120 units have a difference of 208 units between them when 120 degrees apart. Example: a 120/208V, 3 phase system. (120 units to the center (neutral), 208 units between each leg (phase)).
A 347/600V system is a similar 3 phase (grounded wye) system. (600 volts between phases, 347V from each phase to neutral). 277/480V systems is common in U.S. (when there's a neutral).
Note "square root 3" relationship.
