You know that old folk-tale that Eskimos have a hundred words for snow? Turns out that isn't actually true. I grew up in northern Canada and while we certainly dealt with a lot of different kinds of snow, ice, sleet, frost and related phenomena, we still only had a few words. Which turned out to be sufficient.
I don't know enough about the other time-arts like theater and film, but I do know that in music, we have a lot of words for time. Or as it is often put: "timing".
- Time: in music the only time we use the word "time" or "timing" is in calculating the duration of a piece of music. This is the length of a track on a CD in minutes and seconds.
- Tempo: is the word that describes the pace of a piece of music as fast or slow. We actually have a whole bunch of words in different languages that specifically identify a particular pace or tempo. Allegro is a fairly quick tempo and adagio a slow one.
- Beat or Pulse: this is what you follow when you tap your foot or bob your head to a piece of music. If you want to be fancy, you could also call this the tactus.
- Meter: In nearly all music, beats are grouped into packages. When written down, these packages are defined by barlines and at the beginning of the score you are told what is in the packages with a time signature. The time signature tells you the meter, such as 3/4 or 9/8. The 3/4 signature says that there are three beats in a bar and each beat is written as a quarter note. 9/8 is a bit more complicated because each beat is a dotted quarter note and there are three of them in the bar. When you have this kind of thing it is called a "compound" meter.
- Rhythm: is the pattern of notes of different durations that make up the actual melody and accompaniment of the music.
These are the basic words that describe "time" in music. But there are host of other ones that tell the performer to slow down (ritardando, rallentando, allargando) or speed up (accelerando, stringendo) or to modify in some other way the basic pace. Wikipedia has an article on tempo words that lists a lot of examples.
We also use numbers to precisely define an exact tempo and some composers use only that. This is why you may see a track on an album listed as a musical note equals 80 or another number. This means that the note is the pulse and there are X number of them in a minute. In an odd sort of way, that is the "name" of that particular movement. Otherwise it would be allegro or some other tempo word. Interestingly, there seems to be a trend in composers in recent decades to name their pieces with something evocative. So Esa-Pekka Salonen, if he is not writing a concerto, might call a piece Nyx after the Greek goddess of night. Debussy was one of the first to make a practice of this with his Prélude à l'après-midi d'un faune and other pieces. But the tradition of calling a piece "Symphony No. 1" and listing the movements by their tempos still continues with some genres. If Debussy had followed that practice I suppose he might have titled his piece just Très modéré as that is the tempo word at the beginning of the Prélude à l'après-midi d'un faune. But the title is definitely better!
Let's listen to that one to end: