Demisexual is a sexual orientation on the asexual spectrum defined as someone who does not experience sexual attraction until they have formed a deep emotional connection with someone. The connection can be romantic, platonic, or some other form of connection. Forming an emotional bond with someone does not mean that one is automatically attracted to said person, as it just means there's now a possibility for one to feel attraction.
Demisexual can be a sexual orientation on its own or can be combined with other orientations. For example, one could be demisexual and polysexual (demipolysexual), meaning that when one does experience sexual attraction they can experience it towards multiple genders, but not all genders.
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Counterparts to demisexual include demiromantic, demiplatonic, and demialterous.
Edit: Idk who needs to hear this but a Demisexual person CAN get crushes and experience sexual attractions toward people they have just met or even complete strangers at times. It's just that they don't last very long at all and exist mostly as fantasies. They can also have active sexual fantasies about real and fictional people; They just can't be reciprocated in real life without a deep connection.
(Read this comment thread: https://lgbta.wikia.org/wiki/Demisexual?commentId=4400000000000049995)
But, of course, every person experiences different attractions in their own way. And that's totally valid!
Etymology
The prefix demi- means half or partial. The prefix is also used in words such as demigod.