This is a really great question, and I'm thoroughly enjoying reading the responses you've already received. Perhaps I can add a little insight of my own.

First, when it's not the right person, there is a whole series of emotions and moods to go through as the infatuation stage falters. You might go about convincing yourself that the situation is better than it is, or that it might become better with time, or that your idea of the person is somehow more accurate than what your eyes and ears are telling you. This is natural, it's part of the bonding draw that everyone feels. This is also why so many people end up in dead end relationships that go on much longer than they should. I know this from experience. The doubts and annoyances creep in, the desire wanes, the overall enjoyment of their company evaporates. All that you're left with, if you continue to pursue it, is your idea of what you wanted it to be. Eventually you might realize that a lot of pain and misery could have been avoided by simply accepting the other person as they were, and ending it.

And now that that depressing shit is out of the way, I'd rather share what it can be like with the right person.

The immediacy of the lust and desire might soften as the change in your dynamic occurs, but the improving comfort level with the person takes over. Mutual trust and admiration of the other person as a person make a whole ton of new things possible. There can be a tremendous feeling of being safe in being yourself, without pretense, without worry, knowing that the other accepts you. The magic really takes off in that state, and it keeps the passion and desire alive and healthy. It might not have quite the same character of immediate need that it did in the beginning (at least not all the time), but when it turns into a lasting desire to make every experience with your partner the most glorious it can be, it's even better. Someone else mentioned a feeling of great calmness, and that is definitely an aspect, but it's brought about by the combination of all the things I just listed. In the best of circumstances, both people start to feel that the other is simply an extension of themself.

The annoyances become trivial, the aesthetic imperfections become unimportant, the thing that really matters is the mutual acceptance of the other as a whole person.

While this kind of progression might be common or even typical (in either the right or wrong person context) it isn't always the case. There is a ton of variation in the order of events and feelings, and the circumstances that lead up to the relationship play a vital role as well. I consider myself one of the lucky ones. The right person for me, I've known for a decade now. I've known that she was the right one for more than half of that time. It was very hard at times. We went from friends, to beloved friends, to a romantic interest, to some time apart, to finally come together again and accept that it's time to make it happen. It's not typical, and it's not even linear. It should still be said that a solid friendship first can be one of the best foundations for a really good long term relationship, especially if you're shooting for a life long coupling.

This turned out a lot longer than I intended. I guess if I were to try to TL;DR it, I would say the most important thing is to have and recognize a true admiration and interest in your partner as a whole person before getting swept up in that crazy OD of brain chemicals that we call infatuation. At least if permanence is your goal.