Meaning of every other day/week [duplicate]

Every day means: day1, day2, day3, day4, day5, etc., i.e. every single day, not skipping any.

Every other day means: day1, day3, day5, day7, etc., i.e. skipping every second day - one day on, one day off, and so on.

The word other, in this case, is used to mean alternate. It is similar to saying "every odd week" or "every even week", only it doesn't matter what number week it is, it only matters that it's every second one.

Saying "every week" and saying "every other week" are not equivalent; it has nothing to do with rules of grammar, only with the fact that it's every second week.