They are grossly simplified and considered in a shallow manner.
There are 10 "Zodiac signs" you can have in your chart, mainly, based on the placement of your 10 planets. Generally, only the Sun to Saturn are important, while Uranus to Pluto are generational and have a lesser influence on the individual, except when they are prominent in the chart. Some planets may also be in the same sign.
What these consider is only your Sun sign, which is the placement of the Sun when you were born. What they do for considering affinity is, again, a gross oversimplification that holds little to no value: they consider the entire Sun sign as if the Sun occupied the whole instead of a specific degree in the sign, and then they consider the whole signs for everything else drawing a whole-sign aspect between your Sun sign and the other person Sun sign.
For example, if you are a Gemini sign, they will tell you are compatible with any sign that is in a trine position to that (Libra and Aquarius) and those who are sextile to it (Aries and Leo). A trine aspect is usually when energy work in harmony between them, while a sextile aspect is usually when energies support each other.
Some sites will also go by the maxim "Opposites attract" and will tell you Gemini is compatible with Sagittarius, which may or may not be true if your Suns were actually in opposition, because opposition can be good, bad or neutral depending on how the energies are handles.
Some sites will also consider that "Like attracts like" and will tell you that Gemini people are compatible with one another, but this is again an oversimplification. Similarity may as well end in ego clashes. Not to mention, many other factors they do not consider (if there is even a real conjunction) determine how the energies of Gemini are actually expressed in your personality. Then there is a thing called maturity. Each aspect of Gemini like any sign can be expressed according to your level of maturity on that aspect. So in some ways one Gemini person may be more immature than the other, and vice versa.
If this is too cryptic, just read the JoS.
But, yes, consumer astrology is mostly useless.