(Unless we're cooking meat. Then we sear the meat first, remove it from the pan, and then start cooking the onions and everything else.)
To cook them, we put a pan over high heat and add a dribble of olive oil. When the pan and oil are both hot, we turn the heat down to medium and add the onions. We stir them occasionally to make sure they cook evenly and aren't burning, but mostly leave them alone to do their thing.
After a few minutes, the onions will turn translucent and lose their crispness. A bit of browning is fine, but the heat should be low enough to cook the onions without giving them a lot of color. If we're cooking other vegetables, we add the next one now.
This is the technique we use most often when cooking onions. It creates a solid, non-assertive flavor base, and as we continue to cook, the onions literally melt into the dish.
If we're doing a stir-fry or something where we want a more assertive onion flavor, we cook the onions quickly over a higher heat. The onions will develop caramelized brown spots and a bit of translucence around the edges, but retain their crispness and a slight raw bite.
And of course, cooking onions over a very low heat gives us yet another flavor transformation. During the long cooking over low heat, the onions lose all of that astringent bite and become completely, deliciously caramelized. If you've never had caramelized onions, try cooking them this way at least once. You'll be amazed at how sweet and tender the onions become!
Is this how you also cook onions, or do you follow a different technique?