"It doesn't matter what is on the outside. It is what is on the inside that matters most."
This brilliant age old advice stands the test of time. It is a testament to the importance of judging personality and character in a person, rather than simple physical attributes like age, race, or attractiveness. However, as good as the advice is, it doesn't quite work when applied to relationships.
"Beauty is in the eye of beholder."
This bit of Shakespearean insight is much better relationship advice. It doesn't matter what society considers to be traditional beauty, intelligence, or wit, it only matters whether you find those particular aspects of your partner attractive.
Thus, the question: How important is physical attraction?
The short answer is: It is reasonably important.
Attraction is Always Important
The long answer is a bit more complicated. The reason that physical attraction is reasonably important is because every form of attraction is important for a healthy relationship. Successful relationships just about require that you be attracted to every aspect of your partner. You should be attracted to their personality, their style, their level of intelligence, the way they smell, the sound of their voice, the way they engage in intimacy, and yes, their physical looks.
If any of these things, or other things about your partner, actively disgust you, turn you off, or lessen your ardour for your partner, it is unlikely that you will be able to maintain a long term, healthy relationship. That one problem will irritate you like a pebble in your shoe. And while some "pebbles" can be removed, physical looks simply won't change any time soon, short of plastic surgery (which should never be the pre-requisite for a relationship).
One important thing to note is that there exists a middle ground between "attractive" and "unattractive." It is possible to neither be attracted to, nor to be turned off by, someone's physical features. Due to the emphasis that people tend to put on physical looks, this is actually rather uncommon. Most likely, the people in your life that you don't find either attractive or unattractive are family members.
That said, it is very possible that you will find yourself deeply attracted to non-physical aspects of an individual, while being almost entirely blase about their physical features. In such a situation, physical attraction doesn't really matter. Your lack of attraction isn't an irritant. As long as the other attractions you feel create consistently warm feelings for the individual, either when you see them or think about them, or both, your relationship shouldn't be harmed by the lack of physical attraction.
When discussing physical attraction, it is important to realise that there is another form of physical attraction, beyond visually enjoy the physical features of someone.
Have you ever experienced that perfect kiss? Or much worse, have you ever experienced that awful kiss?
This is also a form of physical attraction. Enjoying or disliking the way your partner kisses you, has sex with you, or engages in other forms of intimacy, is a form of physical attraction. And this form of physical attraction is probably the most important. If engaging in intimate acts with your partner actively turns you off, the relationship is in a lot of trouble. Your partner may be able to learn to kiss better or be better in bed, but that can be an awkward conversation and a difficult process, especially if the very act of practicing is turning you off.
Handling a Lack of Attraction
If you are physically attracted to your partner, congratulations, you won the lottery. If you aren't, though, it can be rather awkward to tell your partner.
Telling your partner you aren't attracted to them because they are mean or like rugby too much or have terrible taste in style is relatively easy. These are choices the other person made. However, physical looks, and to a lesser degree, skill in intimacy, is something a person doesn't have a choice in. For this reason, society often frowns on people for dismissing others for these reasons alone.
Society isn't entirely wrong, but that doesn't mean you shouldn't be fair to yourself. It does mean that you should be polite and gentle if those are the reasons you aren't attracted to someone. Be honest without being rude and never use insulting words like "ugly", "fat", etc.