Many people imagine that the astrological compatibility between two persons is a matter of knowing which sign is each one and then find if those 2 signs are compatible or not.
Well, the truth is that it is much more complex than that, just think that all the people born during a full chinese year are the same sign, let's say "dragons", so it would be too naive to think that they (many millions) are "more or less" the same, of course they use to share some common characteristics, but not in a uniform or very predictable way.
It happens that the "sign you are" is just the beginning of the story, then it follows the signs belonging to the time, day and month you were born, the 4 "paths" of your natal charts, your elements... that's all I talked about here.
So now we are talking about millions of people belonging to the same sign but with millions of different natal charts, which make much more sense from a mathematical point of view.
There are 2 useful tools when looking at the astrological compatibility of two persons, one is the signs wheel, and the other the life chart.
The signs wheel is just a circle where the 12 signs are disposed orderly so it's easier to study the relationships between them, it looks like this:
There would be many things to say about this, but the most important one is when a star is formed, or almost formed, with the signs of your natal chart.
This is important when studying the astrological compatibility because people who need one sign to complete one of their stars will typically feel attracted to people providing that sign, and very often the feeling is reciprocal, it's like one having the need to give something and the other need to be given that.
On the other hand, the life chart gives an outlook of the energies behind a person and how his/her life is likely to develop, it looks like this:
First thing here is to see which elements are absent from the chart, this is very important regarding astrological compatibility, since people lacking one or more elements will typically complement well with people providing those elements, the best of it is when the two have some element that the other needs, for example, one having plenty of earth but no wood and the other having plenty of wood but no earth.
Then, it's also important to have a look to the different life stages on the chart, people with difficult stages will match better with people that during those years have easier stages, specially when the other person provides the element that is absent during that stage.
So, as you see, it's not a matter of just seeing how well the sign matches but of studying it in much more depth.