Quant is short for quantitative analyst: a person who does mathematical analysis in finance. This node is on the job of quant, what he or she does, and how one could become one.
As mentioned, a quant is an applied mathematician that applies his math to problems in finance. However, money and numbers are closely related, so what makes one person working in finance a quant and the other not? Perhaps the best criterion here is the level of math involved. Quants normally deal with problems that are too hard for their colleagues to deal with, and their colleagues normally have a Bachelors or Masters degree in economics or econometrics. The math tends to be similar to that in applied physics: differential equations, statistics, and a lot of calculus and algebra.
At this point, the similarities between quants more or less end. You see, there are many places in a bank or trading firm where a quant can help out. Some quants work at jobs that are very similar to that of a trader: they develop and test trading strategies. Some even trade themselves, further blurring the line with traders.
Other quants develop models for complex products, such as combinations of bonds, options or other derivatives that have a complex payoff. And yes, the people responsible for building the toxic mortgage bombs that were instrumental in causing the 2008 financial crisis were quants. Financial modeling is not nearly as exact as modeling in, say, physics. Still, one can easily spend as much time on such a model as on modeling, say, a jet engine - and in fact, messing up the financial model could be even more costly than having to recall a few dozen planes.
As such, it is difficult to describe in more detail what a quant's job will entail. Some quants do a lot of data mining, others use analytical methods. Some talk directly to brokers, others work in the backoffice. Some work trader-like 60 to 80-hour work weeks, others work 40 hours in a busy week. Some are aggressive and ambitious, like traders are, others are more scholarly. Some are programming wizards, others struggle with Excel.
How to become a quant
So, you like math and you like trading, and you think you might want to become a quant. How do you do this? Well, first of all, nowadays, there exist dedicated master-level programmes to do mathematical finance. The thing here is that most the really good information is not in academia: cutting-edge knowledge in finance is closely guarded by the companies in the field. As such, most firms will train you on the job.
The main thing one needs to the table is a high IQ and skill with numbers. As far as I know, very aggressive selection on candidates scoring high in thisIQ, or whatever proxy they choose is very common. A score that could get you in at (almost) any other company might not be enough. It doesn't hurt to try at multiple firms, either. Quants are normally hired fresh out of university, and normally have a degree in a field like physics, mathematics or engineering. Some people go in straight after their bachelor's degree, others wait until they have a master's degree or PhD. There is value added in having a better education, especially if one is going to be more involved in deeper analysis and less so in trading.
A quant is a mathematician who works in finance. This is a necessarily vague description, as there are many different types of jobs under the label quant, from scholarly modeling of complex derivatives to developing trading strategies. Becoming a quant in general requires excellent skills in a field that involves a lot of applied math, and the ability to stomach the culture at a bank or trading firm.