Quantitative analysts ("quants") are employed across the financial industry: in investment and corporate banks, ratings firms, brokerage firms, financial data providers and software companies, insurance companies, asset management firms, treasury and finance departments of major corporations, hedge funds, etc.
Some of the positions they may occupy in a company:
front office/desk quant
research quant
quant developer
statistical arbitrage quant
capital quant
portfolio quant (quantitative portfolio manager).
In his article On Becoming A Quant, Mark Joshi summarizes job duties for each category of quants.
A desk quant implements pricing models directly used by traders. This job is very close to the action, on some trading desks it may even involve actual trading.
A research quant tries to invent new pricing approaches or improve the existing ones. Some of them may work on designing new financial instruments, and carry out original research.
A quant developer spends most of his/her workday coding: implementing trading systems or pricing frameworks. Some quant developers work for specialized software companies. C/C++ is still the programming language of choice in the financial industry, but other languages and platforms (Python, R, Matlab) are strongly represented as well.
A statistical arbitrage quant works on finding patterns in data to suggest profitable trades. This sort of job involves a lot of statistical analysis / data science, and is most commonly found in hedge funds or proprietary trading desks. The time horizon of analysis ranges from weeks and months (conventional StatArb ) to seconds and milliseconds (high-frequency trading).
A capital quant works on modeling the bank’s credit exposures and capital requirements. Accounting companies, insurance companies and government regulators also employ people in similar positions.
A portfolio quant uses the optimization techniques to manage the firm's portfolios and optimize the trading costs. Risk management is often an important part of this job.
There are also many other jobs in finance which are not "purely" quantitative, but require considerable mathematical and/or programming expertise. Among those: quantitative structurers (create and test new products based on financial derivatives), quantitative customer support (tailor the financial products to the customer needs and address the customer concerns) and even sales support (create ideas for the sales teams, and explain the new products to the potential customers).
Books to read about quants
My Life as a Quant: Reflections on Physics and Finance
The Complete Guide to Capital Markets for Quantitative Professionals
How I Became a Quant: Insights from 25 of Wall Street's Elite