There are two general meanings.
1. The absence of a sexuality, most notably in humans. Some may just not be sexuallly attracted to others, otehrs may completely lack a sex drive or sexual desires. Common misconceptions are that such people have suffered sexual abuse, are emotionally frigid, are closet homosexuals, masturbate ten times a day, are too ugly to get in a sexual relationship or are impotent. This, more often than not, does not apply.
2. Without sexual intercourse. For example, cells (be it nerve cells, bacteria or other protozoa such as amoeba) reproduce via mitosis (or dividing) as opposed to sexual reproduction. The result tends to be two cells of exactly the same genetic structure as the 'parent cell', though genetic mutation can occur if the DNA is duplicated incorrectly.
Connected words/phrases: asexuality, sexuality, sexual, mitosis, cell division, queer (asexuality is considered a queer sexuality. This is not a negatiive connotation, only meaning 'different').
1. I am not a closet homosexual, I am not impotent, I have not been sexually abused, I am not emotionally frigid, I don't masturbate ten times a day, and while I am one ugly bastard peopel round here are so easy I could get nookie the second I want it. I just don't want it. I am asexual.
2. Apart from rare cases where cells reproduce via meiosis (splitting into four cells), most cells reproduce via mitosis. Originally the human body is just one zygote cell, two gamete cells fused together. At adulthood, there are millions or perhaps billions of cells all with different purposes, nerves, muscle, skin, brain tissue... This is all possible due to mitosis, the non-sexual reproduction of the cells themselves. However, the human body requires sexual reproduction for the initial cell. An example without this is bacteria, which have no form of intercourse and simply divide. This is asexual reproduction.