Aromantic is a term used to define one's romantic orientation (sometimes referred to as a person's affectional orientation), used for those who do not feel an urge to engage in romantic relationships. It is sometimes abbreviated aro.
Aromantic is, for the most part, a modifier used by the asexual community. Being asexual does not necessarily mean that you do not desire romance, but it does mean that terms like gay, lesbian, and especially bisexual may not quite fit. So instead you may hear the terms homoromantic asexual, heteroromantic asexual, biromantic asexual, and panromantic asexual. And, of course, you might also come across people who identify as an aromantic asexual.
The term is not always limited to the asexual community. There are also those who identify as, for example, an aromantic bisexual. And you may have noticed that there is a over-hyped stereotype in Western culture of the 'aromantic hetrosexual male' (although that term is never used in the sitcoms). However, the general population usually ignores subtle flavors of romantic orientation -- even more so than sexuality or gender.
These terms are still fluid, and likely to remain so for the foreseeable future, but it is useful to remember that 'romance' doesn't doesn't just mean roses and candle-lit dinners, but also kissing, cuddling, and even just invasion of personal space. What exactly an aromantic person might desire, or tolerate, will vary greatly among individuals.
As with asexuality, aromanticism is much more likely to be seen as a medical/psychological problem than are other sexual and gender differences, even among the generally liberal LGBTQ population. Not only is this an isolating gender identity, it is one that many people will see as one that a person has chosen for themselves, and therefore should be able to 'get over' or live with without complaint. This is unfortunate, because an aromantic asexual may still desire long-term companionship, children, or other trappings usually associated with romance and sex.
It should be emphasised that identities like quirkyalone and genderless do not have any bearing on being aromantic. Quirkyalone might be compared to gray-romantic, a term for decreased romantic desire, but even that is iffy. However, these terms are generally defined by those that apply them to themselves, so when it doubt, ask questions.