Synastry: your two charts
Sun sign compatibility compares two boxes out of twelve. This is something else: both charts are calculated in full and the ten bodies of one are crossed against the ten of the other, to see what angles they make between them. That is where you actually see what holds you together and what rubs.
The positions are astronomy — the same formulas as the ephemerides — and without a birth time they are calculated for midday, so the Moon can slip into the next sign. The reading is astrological tradition: a way of looking at the relationship, not a verdict on it.
"Are we compatible?" is one of those questions almost every website answers with a percentage. You will not find one here, and not for want of trying: nobody knows how to turn two birth charts into a number. What can be done, and it is a great deal more interesting, is to set the two charts face to face and look at the angles they make between them.
That is synastry. It does not compare your star signs — those are two boxes out of twelve, and you already have that in zodiac sign compatibility — but the ten bodies of one chart against the ten of the other: a hundred crossings, of which the ones falling within orb are kept. That is where the contacts tradition has been watching for centuries show up: one person's Sun with the other's Moon, Venus with Mars, Saturn with whatever it happens to land on.
And something worth saying before you start: the so-called hard aspects are not the bad ones. The square and the opposition are the points where you push each other, where one of you knocks the other out of place. A relationship made only of trines is comfortable and falls asleep; the one with friction keeps moving. What matters is not how many there are but knowing what they are about. If you have not got yours yet, start with your birth chart: both are calculated here with the same engine.