Answer: Before there were metronomes - which means, before the time of Beethoven - musicians had to just rely on descriptive words like "Allegro," "Andante," "Adagio," etc. It was not a precise thing, so there wasn't really a "right" tempo, though there could be a wrong one if you were doing something really crazy.

Some of the old terms have an approximate relation to real human experience. "Andante" means "walking speed," for example, though there may be questions about which note value is doing the walking. Some uses of "Andante" seem to refer to the speed of even eighth notes in the bass rather than to the speed of the quarter notes. These tempo indications also will sometimes indicate mood: "Allegro" we generally take to mean "fast," but literally it means "cheerful" or "merry" or "lively." "Adagio" does literally mean "slowly," but also "gently." Sometimes other terms are added in an attempt to make things clearer, as in Beethoven's famous "Allegro ma non troppo," meaning "fast, but not too much." But that kind of tosses it back in your lap again - what's "too much?"

Add to that that even metronome marks are not necessarily "right." Beethoven was very pleased by the invention of the metronome and used it to set down markings for each of his symphonies. But there remains a lot of controversy about whether the marks he was reported as making were always what was intended - for instance, whether a marking referred to the speed of a quarter note or the speed of a half note, etc. Similarly, the Largo of Vivaldi's Winter concerto, in 4/4, has been taken by one well-known conductor to apply to the half note rather than the quarter note, so that only the harmony changes are slow.

You want a shorter answer? Be guided in a rough way by the tempo marking, generally thinking of Allegro 4/4, for example, as having quarter notes that are perhaps the speed of a power walk. In "cut time," or 2/2, the music looks just like 4/4 but now you're power walking with the half notes rather than the quarter notes. Andante could be more of a relaxed stroll. Adagio would mean your walk would be solemn, like a funeral march. But most of all, play so it "sounds right" to your own ears, and at a tempo you can handle without too many errors. The pianist Glenn Gould played Mozart's Sonata in A much, much slower than anyone ever had before, so far as we know - but was it wrong? People will disagree, but many think it sounded very good. He made it sound right.