Alimentary is an adjective that refers to a level of gradation. The word was invented to cover off cases where someone awkwardly introduces a new level that falls between two existing levels that have inconveniently already been named elementary (or primary) and secondary.
For example, middle schools teach the grades between elementary (or primary) schools and secondary schools. It would thus be appropriate to call middle schools alimentary schools.
Some people have trouble digesting the concept of the word alimentary because, they say, it would make much more sense to just rename the existing levels so the new level could be included without having to resort to the fudge of alimentary. They would, for example, suggest that middle schools should be renamed secondary schools and secondary schools should be renamed tertiary schools. However, this would cause far to much confusion because the elementary and secondary labels are already firmly entrenched, at least in the case. Imagine just the cost of changing the signs on the schools. It would be financially infeasible.
Besides, colleges and universities are already referred to as tertiary education, so they would have to be renamed too. And who the hell knows what the word for the fourth ordinal level is? It would all be too much for most people.
So, the word alimentary comes in very handy at times. Of course, all hell breaks loose if some damned fool introduces, not one, but two new intermediate levels between elementary and secondary.