We've had an inquiry about the difference between infirmary and infirmity. As in so many cases, the Romans are most directly to blame for the similarity between these words. Let's explore a little farther with the help of the staff and faculty of Hogwarts, from the Harry Potter series.
In Latin, firmus means much the same as firm does in English: firm, steadfast, solid. The Latin prefix in- means not or without, so infirm means much the opposite of firm: that is, weak or ill. Both infirmary and infirmity come from this word, but with different suffixes, so in fact we have here a great example of how English builds words with Latin roots, prefixes, and suffixes.
Webster's (that is, the Merriam-Webster Dictionary) tells us that infirmity means "the quality or state of being infirm" as well as "disease, malady." In contrast, an infirmary is "a place where the infirm or sick are lodged for care and treatment." We can guess that the differences in the definitions must be due to the suffixes.
The suffix -ity comes from the Latin suffix -itas, usually by way of medieval French, where it became -ité. This ending takes an adjective and makes it into a (usually) abstract noun that means the state, quality, property, or condition corresponding to that adjective:
Professor "Mad-Eye" Moody was such an imposing and alarming figure that it was hard to regard his missing eye as an infirmity.
On the other hand, the suffix -ary is (according to the Online Etymology Dictionary), from the Latin suffix -arius/-aria/-arium (depending on gender - m/f/n), which implies "connected with or pertaining to." Most of the English nouns that use this ending refer to actual places or people:
Madam Pomfrey could always count on having most of the beds in her infirmary occupied after an important Quidditch match..
To remember which word is which, perhaps it might be useful to remember some of the other words that use these endings. The Hogwarts infirmary likely also serves as a dispensary, which you might find useful after an unpleasant encounter with a Quidditch adversary. However, to overcome the adversity of a serious infirmity that might affect some ability in the long term, you might require a stint in St. Mungo's Hospital.
Word Trivia: You'll note that adversary and adversity form another pair of nouns that are related in the same fashion as infirmary and infirmity.