"1 and 2 and 3 and 4."
If beats 1 and 3 are strong (aka on-beat) and 2 and 4 are weak (aka off-beat). What is the 'and' called? I'm wondering about the terminology.
If you are talking to other musicians about what is or should be played you might say "on the upbeats" "on the off beats" or "on the up strokes". Context is key here as the meaning can easily be confused with weak beats. You could even clarify by saying "the 'ands'" but this could sound a bit amateur so use sparingly.
If it's called anything, it's called the 'and'. But it isn't really called anything special.
It's a timing mnemnonic. Here are several common ones, for the sake of simplicity, assume 4/4 time.
Eighth notes: One and Two and Three and Four and
Triplets: One and a Two and a Three and a Four and a
Sixteenths: One ee and a Two ee and a Three ee and a Four ee and a
Fives: Shoot the composer. :-)
I'm not aware of a good term for this, but one may look at the idea of secondary stress in language for a useful analogy. In that model, we have three levels of strength: primary stress, secondary stress, and unstressed. However you look at it, the "strength" of a note is not binary; there is a strong-weak spectrum.
Another factor could be thought of as "zoom," by which I mean the level of subdivision. In a 4/4 measure full of sixteenth notes, you will have a different analysis compared to a 4/4 measure comprising two half notes. In the latter case, of course, the second half note is weaker than the first. In the former case. it may or may not be desirable to distinguish the ninth sixteenth note from the first.
So in your example, which implies eighth notes, you might quantify the stress with four levels, thus:
Meter: 1 & 2 & 3 & 4 & Stress: 1 4 3 4 2 4 3 4
Here, you might say that the offbeats are unstressed, beats two and four have tertiary stress, beat three has secondary stress, and beat one has primary stress.
One of my professors liked to use the phrase "elevator operator" to illustrate this. It has eight syllables. The first syllable of each word is the primary stress, and the third carries the secondary stress. But the primary stress of the first word will normally be stronger than the primary stress of the second word. There could also be contexts in which the second word is more important, for example when distinguishing "elevator operator" from "elevator mechanic."
Finally, a side note on the nature of musical stress: it is not necessarily expressed through volume alone, or even at all. It can also be realized by lengthening and shortening beats or by articulation. A great performer does a little of each of these. "Strong" and "weak" are often taken to mean "in volume," but it's better to think of these terms as shorthand for something more subtle. In the baroque period, the distinction was commonly between "good" and "bad" instead.
Focusing on terminology, which appears to be the crux of OP question:
Jazz musicians will often talk about the "and" of a particular beat. For example, one might say "that phrase starts on the and of the four," or "come in on the and of the two."
But, I am unaware of any common and accepted terminology that refers to the "ands" within a measure as a general class of beats.
With 1 “and” 2 ... we’re only counting the beats and not adding them. The 1234 is the pulse, or breathing in, the “and” is the breathing out. In once counting very slowly you will agree. Instead of “and” you can choose any syllables or vocals a or o or e ...
Music and rhythm is coming out of our body. It’s like as you were rowing, jumping, sewing, or hammering a nail...
So the “and” doesn’t stand for + (plus), that says: it and doesn’t mean to add like 1 and 2 = 3
(in this case a 4/4 measure would be 10). However for counting in music you can also notate 1 + 2 + 3 + 4 + instead of 1-a-2-a-3-a-4-a.
(by the way: Kodaly also had a kind of rhythmic language for different rhythmic patterns)
as far as I understood the weak beats in a measure are known as backbeats and the latter half of a single pulse if the off beat so the ands in your case are the off-beats and the 2 and the 4 of a 4/4 bar should be backbeats.
I am not a English native, but this is what I have known so far. So for a 4/4 measure
1 and 2 and 3 and 4 and one and two and three and four and 1 2 3 4 beat offbeat backbeat offbeat beat offbeat backbeat offbeat
there is no real reason to explicitly call the backbeats as so, they are still bits just to emphasize the 4/4 in this case.
I would call those the upbeats. (And it doesn't have to be 4/4, could easily be 3/4 and probably any regular time signature.). If you're tapping your foot on the beat, it's when your toes go up between beats.
I think I would know what you meant if you said offbeat, but I wouldn't use that myself because if you want something off-beat that doesn't necessarily mean exactly halfway through the beat, which upbeat does actually imply.
Confirming @David Bowling, yes in jazz it's common to say something like "the & of three" to refer to the '&' between 3 and 4. But I've only heard that in the singular instances, not generically to all instances. No one says "hit cowbell on all the &s". I think it'd be "hit cowbell on all the upbeats".
Also, this is regardless of whether you're playing a rock song (big stress on 1st beat, medium stress on 3rd beat, and light on 2 and 4) or a jazz tune (medium stress on 2 and 4, light on 1 and 3).
Hope that all makes sense.