Demisexual means only feeling sexual attraction to someone after a close emotional bond has formed. Not preferring that, or choosing to wait for it. The attraction is simply not there until the bond is, and no amount of time or good looks produces it on its own.
Why it is not the same as taking things slowly
This is the usual objection and it misses the point. Plenty of people feel attracted to a stranger and choose not to act on it until they know them. That is a decision about behaviour. A demisexual person does not have the attraction to act on in the first place, and it appears later or not at all.
The practical difference shows up in ordinary moments. Walking past someone striking and feeling nothing. Being told a celebrity is attractive and being able to see it as a fact without feeling it.
Where it sits
Demisexuality is usually placed on the asexual spectrum, between asexual, meaning little or no sexual attraction, and allosexual, meaning the usual experience of it. Someone demisexual does experience attraction, under a condition.
It says nothing about gender. A demisexual person may be straight, gay, bisexual or pansexual as well, because that is a separate question.
Demiromantic
The same pattern applied to romantic rather than sexual attraction. The two can run together or separately, which is why people sometimes use both words or only one.
The argument about it
Some people say this is just how attraction works for everybody and that the word medicalises something ordinary. Those who use the word say the difference is not degree but kind, and that having a word for it ended years of assuming something was wrong with them. Both are worth hearing. The word costs nothing and clearly helps people describe themselves, which is most of what a word is for.
Why it appears in adverts
Rarely, because paid bookings are arranged around time rather than bonds, and the arrangement does not depend on attraction forming. Where orientation labels do appear on profiles they describe who someone will see. The glossary covers the labels that turn up more often.