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Attraction in the context of AVEN refers to a mental or emotional force that draws people together. Asexuals do not experience sexual attraction, but some feel other types of attraction. There is some amount of debate as to what types of attraction actually exist.
Disclaimer: not everyone agrees with this definition of sexual attraction
Main article: Sexual attraction
Sexual attraction is a feeling that sexual people get that causes them to desire sexual contact with a specific other person. It is often, but not always, felt along with other forms of attraction - i.e. sometimes a person experiencing sexual attraction will only want sex, such as some friends with benefits relationships, and other times they will desire sex as well as romantic interaction or other things.
Sometimes asexuals will desire sexual contact for other reasons besides attraction (e.g. in order to make a sexual partner happy, to satisfy a curiosity, to have a child, or to prove to themselves or others that they are "normal"). It therefore becomes difficult to define sexual attraction exactly, as it is not considered by many asexuals to be the same as desire. One good rule of thumb is that sexual attraction involves a desire for the sexual act itself, rather than its social consequences. Some models of asexuality, make distinctions between different kinds of sexual desire, and allow for asexuals to feel some varieties but not others.
Sexual attraction is not the same thing as sex drive, although in sexuals the two often go together. When asexuals experience sex drive, it is not connected to attraction or desire, and can thus be taken care of by oneself.
Main article: Romantic attraction
Romantic attraction is a feeling that causes people to desire a romantic relationship with a specific other person. Many asexual people experience romantic attraction even though they do not feel sexual attraction. Sometimes this romantic attraction is directed towards a specific gender, giving asexuals who experience it a "romantic orientation" that is different from their sexual orientation. Other asexual people do not feel romantic attraction, and classify themselves as aromantic as well as asexual. As mentioned in the Aromantic FAQ, not all aromantic people are asexual; there is as much variance in their sexual orientations as there is among people who experience romantic attraction. (For example, one bisexual aromantic makes comics like this.)
What exactly constitutes a romantic relationship or romantic attraction is difficult to define, and some asexuals reject the romantic/aromantic dichotomy altogether. Some define a person's approach to relationships as partner- or community-based. Partner-based intimacy takes place between an exclusive pair of people, whether or not this pair of people is sexual or traditionally "romantic". Community-based intimacy takes place between a group of more than two people. People who depend on community-based intimacy do not see a need to pair off into couples, but this does not necessarily mean that they are less capable of forming strong emotional connections with others. 
Crushes and Squishes
A crush is a romantic attraction to someone, a desire for a romantic relationship of some kind, a desire that is possibly temporary in nature, possibly never to be acted upon. A squish is an aromantic crush, a desire for a strong platonic relationship with someone; this envisioned relationship is usually more emotional intimate than a typical friendship.
There is a fine line between a crush and a squish. Both crushes and squishes could involve persistent thoughts about the person of interest, self-consciousness around that person, desires to be with him or her, fantasies about physical (not necessarily sexual) contact with him or her, or any combination of these. However, crushes sometimes entail jealousy of partners of the person of interest, and desire for romantic contact (such as kissing), a dating relationship, or marriage, while squishes do not.
Some asexuals report a desire to do sensual (but not sexual) things with certain people, especially relating to tactile sensuality such as cuddling. This experience can be classified as sensual attraction. Some asexuals are uncomfortable with this classification, since they can also get sensual pleasure from nonhuman objects such as pillows, paintings, or pets, to which they do not consider themselves "attracted". It is also sometimes difficult to fully distinguish sensual things from sexual ones in a relationship, especially between sexual people.
Some asexuals report feeling an attraction to other people that is not connected to a desire to do anything with them, either sexually or romantically. They simply appreciate their appearance. This is called aesthetic attraction because it is thought to be similar to other aesthetic desires, such as the desire to keep listening to a good song or to keep looking at a beautiful sunset.
It is possible to define aesthetic attraction as a subset of sensual attraction, since the act of observing a person's appearance or behavior can be thought of as pleasure involving the sense of vision or sometimes hearing. Some asexuals reject the concept of aesthetic attraction for the same reason that they reject the concept of sensual attraction - i.e. it feels the same to them with people as it does to inanimate objects and other things that they do not generally think of themselves as being attracted to.