The second definition in JoeBright's answer is correct: thot is an acronym that stands for "That Ho Over There." The word has a negative connotation and is such sometimes used as an affectionate insult to close friends, but is almost always used to describe women. Ho is commonly misspelled as hoe, but intended as a short form for whore. It's generally considered AAVE, with multiple examples in hip-hop culture.
Unlike in the case of YOLO, where Drake's song "The Motto" popularised it, there seems to be no single event. Rather, the word rose to popularity gradually.
This is going to be long...
I did find a song called "Thot Walk," performed by artists John Boy, Kayleb, and Yung Cyph, but it was made after thot reached its peak popularity on Twitter. (It did start a trend on Vine, but that's another story.)
Examples of characteristics the song states are typically attributed to a thot: (couldn't find lyrics, all by ear):
0:32 "Showing off her body on her Instagram / just to get a hundred Likes on her picture"
So desperate for attention is this thot that she dresses as provocatively as possible to attract as many Likes as possible, most of which come from heterosexual men (for reasons which should be obvious).
0:40 "Got no face, but all body"
She has a body considered by society to be physically attractive (large breasts & butt, most likely) yet her face is unattractive. This may be because she has an awkward bone structure, is unkempt with her appearance, or is unskilled at applying makeup in a flattering fashion.
0:43 "Go it girl, twerk it, make a movie / If you ain't going to do it for me, do it for the Vine"
It's common on Vine for users to share videos of themselves or others twerking. Since some can even experience their 15 minutes of viral video fame as a result of a twerk Vine, the phrase do it for the Vine has leverage in persuading someone to do anything they wouldn't normally do (not just twerking) for the sake of humour or interest. I believe, however, that Kayleb is using the phrase with thinly-veiled condescension, thinking of girls who twerk for Vines as gullible or attention-seeking.
2:26 "If you one of the thots in the VIP / You'd have fucked the promoter, you don't need no ID"
Those in the VIP section of a club are there most likely either because they have money, fame, or connections to someone associated with the club. John Boy implies that having sexual relations with the promoter is a cheap or dishonest way of getting in.
2:32 "She act like a hooker with a cup full of liquor"
Once under the influence of alcohol, a thot will become promiscuous (or exchange sexual services for money, I wasn't sure).
Along with the concrete examples of thot behaviour, these two lines may also be of interest.
1:21 "She a thot so she talk like one / She a thot so she walk like one"
2:20 "You can tell she a thot by the way that she stands"
These two phrases, both without followup, suggest that there is more to being a thot than just simply being promiscuous. Part of being a thot has to do with how one carries oneself - perhaps speech patterns, lexicon, gait, posture, and/or body language are all variables in determining whether someone is a thot.
I've also searched up the word on Twitter to see how it'd be used in context, and while I did see a few instances of it being used as a shortening of thought as GEdgar suggested, those were well in the minority.
It is illogical to discard a diamond and replace it with a rock because of the drop in value: by analogy, a thot has much lower value than a wife.
Birds of a feather flock together.
Thot was clearly being used as an untrue allegation to save face publicly.
Also provides evidence to suggest that a thot carries herself differently. Presumably, a woman who "runs game" could very well also be promiscuous, but carries herself in such a way that earns her respect, reverence, or unrequited love.
I can assume from this tweet that the scenario in which a girl previously perceived as beautiful becomes a thot is common. The user laments because he believes becoming a thot is shameful.
An exception to the rule: Previously used only by men toward women, this is a woman not wanting to associate herself with anything having to do with thots.
As previously stated, a thot is most definitely promiscuous, but considering the evidence, one can start to form a profile.
A thot is well-connected on social media, but most likely has received most of her attention from desperate heterosexual men hoping to get an easy lay. Unattractive with a sense of grandeur, or perhaps attractive only with makeup, she barely only looks good enough to attract this attention. She is shameless, dishonest, or not self-aware, with further personal boundaries than most. There are a wide variety of men with whom she is willing to sustain a sexual relationship, reoccuring or not. She twerks, or does any other activity notorious for being a blatantly sexual way of attracting attention. She is a common enough occurrence for her value to have diminished. Her behaviour would be unsuitable in polite company. She befriends other thots, while other women who see themselves as better do not wish to associate with her.
Also, that ho over there is a phrase that puts physical distance between the narrator and the ho. Extrapolating, it could imply disinterest or detachment, as if the narrator knows there will be another ho over there, and over there, and over there.
And that's a thot.
The social implications of the word and its usage are a whole other can of worms, and out of the scope of this site.