"Convert" is the English word most often used for one who embraces a religion after being raised in another faith. A common definition of the word "convert" is to "change from one religion or belief to another." Among Muslims, you may hear people refer to themselves as "reverts" instead. Some use the two terms interchangeably, while others have strong opinions of which term best describes them.
Revert to What?
The term "revert" stems from the Muslim belief that all people are born with a natural faith in God. According to Islam, children are born with an innate sense of submission to God, which is called the fitrah. Their parents may then raise them in a particular faith community, and they grow up to be Christians, Buddhists, etc.
The Prophet Muhammad once said: "No child is born except upon fitrah (i.e. as a Muslim). It is his parents who make him a Jew or a Christian or a polytheist" (Sahih Muslim).
Some people, then, see their embrace of Islam as a "return" back to this original, pure faith in our Creator.
A common definition of the word "revert" is to "return to a former condition or belief." A revert is returning back to that innate faith that young children innately connect to.
Convert?
There are some Muslims who prefer the term "convert." They feel that it is more familiar to people and causes less confusion. They also feel that it is a stronger term, better to describe the life-changing path that they have chosen. Some didn't have a strong sense of faith as a child, or were raised in an atheist family, so they don't feel that they have anything to "go back" to.
Convert / Revert / Either / Or?
Both terms are commonly used to describe those who embrace Islam as adults, after having been raised in or practicing a different faith system. In common usage, "convert" may be used just because it is more familiar to people, and "revert" is used only amongst Muslims, who understand the usage of the term. Some individuals feel a strong connection to the "return" to their natural faith, though, and prefer to be known as "reverts" by all. Since this is a less familiar term to most people, they may then need to explain what they mean. In writing, some people just write "revert/convert" to cover both positions and not offend anyone. In conversation, people will generally follow the lead of the person who is sharing.
Blessings for Those Who Come to Islam
"Those to whom We sent the Book before this, they do believe in this revelation. And when it is recited to them, they say: 'We believe therein, for it is the Truth from our Lord. Indeed we have been Muslims from before this.' Twice will they be given their reward, for they have persevered, and they avert evil with good, and they spend in charity out of what We have given them" (Quran 28:51-54).