"Attraction" in the context of AVEN means 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.
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) mixed with another form of attraction - i.e. sometimes a person experiencing sexual attraction will only want sex, 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, for example, in order to make another person happy who is sexual, or to prove to themselves that they are "normal". It therefore becomes tricky to define sexual attraction exactly. 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, such as Rabger's, 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.
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. It is speculated that aromantic but sexual people also exist.
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 necessarially mean that they are less capable of forming strong emotional connections with others. Relevant podcast
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. Relevant thread
Sensual attraction is distinct from romantic attraction, because some asexuals desire romantic relationships without sensuality in them, and other asexuals desire sensuality without romantic committment.
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 merely enjoy observing these people. 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 behaviour 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.