Investment Banks
These are probably the most recognisable institutions in this sector eg: HSBC, Goldman Sachs, JPMorgan, Citi, UBS etc. - some of these banks also offer retail banking services. In a nutshell, investment banks provide services to organisations and high net-worth individuals who want to borrow, lend and invest money.
The services they offer are wide-ranging which is why they are seen a popular option for graduates as they have a number of divisions and roles graduates can choose from. Investment Banks have multiple divisions carrying out a range of work eg: Mergers & Acquisitions, Research, Asset Management, Sales & Trading, Technology, Infrastructure, Retail banking, Operations etc. Firms sometimes call the various divisions by different names, so carefully review firms' websites to understand the work each division does.
Retail Banks
These are the more recognisable “high-street” banks that focus on personal and smaller business banking for the public eg: HSBC, Santander, Barclays, Halifax etc. These banks can provide financial services to members of the public and smaller/mid-sized businesses eg: personal banking, loans, credit cards etc. Some retail banks are divisions of larger investment banks.
Asset Management Firms
These firms manage investments on behalf of others eg: pension funds, governments, high net-worth individuals and corporates, to increase their client’s portfolio whilst at the same time mitigating risk. Examples of "stand alone" asset management firms are BlackRock, Fidelity, Baillie Gifford - many investment banks have asset management divisions.
Private Equity Firms
Private Equity (PE) firms earn money by charging management and performance fees from investors in a PE fund. For example a PE fund may acquire a company, improve its performance by restructuring and selling it. The larger firms in this area include: Blackstone, The Carlyle Group and KKR and although some offer graduate entry-level positions and internships, others don’t and people often join these firms after working in investment banking for a few years.
Quantitative/Proprietary Trading Firms
These firms (eg: Optiver, Jane Street, G-Research) are sometimes known as "electronic market-makers". They develop trading strategies based on quantitative analysis using automated trading models and typically hire graduates with Masters and/or DPhils/PhDs in STEM subjects.
Private Wealth Management Firms
Private wealth managers specialise in managing the finances of high net-worth individuals in order to grow their assets. They can work in a wealth management firm, in a division of a bank or independently.
Investopedia has a great dictionary of finance terms which is a very useful introduction to financial language and terms and E-Financial careers has a useful "Student Guide to Careers in Banking & Financial Markets" ,that gives a high-level overview of the sector.