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.