Your Zodiac sign, also called your sun sign, is simply the position of the sun in your horoscope at the moment of your birth. The complete horoscope includes the planets, the astrological Houses, and the aspects (angles) between planets. That's a whole lot more.
Therefore, predictions based only on your Zodiac sun sign can't be very precise. Still, they give some general clues, since the sun is almost as important in the horoscope as it is in the sky.
It shows your situation and sentiment when you're not involved in other things, for example when you relax or are unoccupied by something special. Your natural state.
Now, in your partner relations, being able to relax together is fundamental, so comparing your Zodiac signs has something to say about how compatible the two of you are.
Fathers
Your Zodiac sign is where the sun is in your horoscope, and in astrology the sun reveals your relation to your father. So, comparing Zodiac signs will also tell something about how your fathers relate to you and your partner, as well as to one another.
Synastry
Still, you must be aware that other things in your complete horoscope may very well suggest something different. Certainly, they add to the picture.
So, if you're serious about using astrology to find out the compatibility with your partner, I recommend that you have a look at both your complete horoscopes. That's called synastry.
For a guide to how you make your own horoscope chart (and that of your partner), check here:
Your Own Horoscope.
Anyway, for your amusement – here are a few words about how compatible you and your partner are, based only on your sun signs. Take it for what it is.
Aries Compatibility
Taurus Compatibility
Gemini Compatibility
Cancer Compatibility
Leo Compatibility
Virgo Compatibility
Libra Compatibility
Scorpio Compatibility
Sagittarius Compatibility
Capricorn Compatibility
Aquarius Compatibility
Pisces Compatibility