What is on axis and off axis of a speaker? What controls it? Is one better than the other? Is it measured both horizontal and vertical? Why do some speakers do some do better?
Lol, wow, that is a lot of question on some deep subjects in speaker design. I can only give surface answers here, because I don't have time to write a multi-volume book, lol.
Axis is the angle at which the listener is at with respect to the speaker. If you are standing dead ahead in front of the speaker, that is zero degrees which is called 'on-axis' or the 'direct axis'. If you are standing to the side of the speaker at some angle, that is 'off-axis'. Here is a picture which illustrates the concept:
Looks like the guy in that pic is standing about 45 degrees to the left and right of the direct axis, btw.
What controls the axis response is the design of the speaker. We call the axis response the 'directivity' of the speaker. Speakers that are highly directive do not shoot sound of of a wide frequency spectrum out at a wide angle; they have a very narrow angle at which they project sound, and consequently, they have a small listening area at which they sound good. Low directivity speakers project a wide frequency band of sound out at a very wide angle, so they can sound good at a very broad listening area in front of them.
Neither low nor high directivity speakers are better than the other, it depends on the situation requirements. There are things to consider like the acoustics of the room and the width of the listening area when thinking about the directivity of a speaker.
One thing to keep in mind that while most speakers are designed to be listened on the direct axis, many speakers are designed to be listened at an off-axis angle. If you want to know the best angle to listen to a speaker at, look for manufacture recommendations or off-axis dispersion measurements.
The axis can be measured by both vertical and horizontal, but people mostly mean horizontal in conversation, at least if they don't specify vertical. If you read a comment where someone says, "these speakers have a great off-axis response", they will mean horizontal.
Regarding how well a speaker handles off-axis response, the best design approach is to have a smooth off-axis response that mirrors the on-axis response. The reason for this is that a lot of what people hear in-room is actually acoustic reflections from the room surfaces rather than the direct sound of the speaker. So if a speaker had a perfect direct-axis response but a very erratic and uneven off-axis response, that speaker can still sound terrible.
Here are some examples of good and bad off-axis response. Here is a good on and off axis response, that shows direct axis, 15 degrees, and 30 degrees; they are all beautifully uniform:
Here is a miserable on and off axis response:
Look at that big suckout from 2 kHz to 5kHz at 30 degrees, this speaker is going to have a veiled sound in-room because of that.