This is a financial institution providing services for businesses, organisations and individuals. Services include offering current, deposit and saving accounts as well as giving out loans to businesses. 
A commercial banks is defined as a bank whose main business is deposit-taking and making loans. This contrasts with an investment bank whose main business is securities underwriting, M&A advisory, asset management and securities trading.
Commercial banks make their profits by taking small, short-term, relatively liquid deposits and transforming these into larger, longer maturity loans. This process of asset transformation generates net income for the commercial bank. Note that many commercial banks do investment banking business although the latter is not considered the main business area.
Examples of commercial banks include HSBC and Bank of America Merill Lynch. 
A commercial bank is one primarily engaged in deposit and lending activities to private and corporate clients in wholesale and retail banking. Other services typically include bank and credit cards, private banking, custody and guarantees, cash management and settlement as well as trade finance.
The term gained prominence as a counterpart to “investment bank” after the introduction of the Glass-Steagall-Act (1933) in the USA that separated capital markets business from deposit taking institutes in the aftermath of the great depression.
However, the separation between commercial and investment banking softened more and more during the age of financial liberalisation and globalisation from 1990 to 2007. Banks moved away from low margin deposit and credit business to high margin capital market investments, accompanied with a respective increase in risk.
This was backed by politicians and regulators, who increasingly reduced the influence of the state in the overall economy and, in particular, in the financial markets.
Eventually the Clinton administration in the USA legalised universal banks, which cover both commercial and investment banking activities with the so-called Gramm-Leach-Bliley-Act in 1999, the basis for the typical business model of banks in continental Europe , which repealed key provisions of the Glass-Steagall Act.
After the recent global financial crisis, however, financial regulation has become prominent once again in the US. The Frank-Dodd-Act implemented the Volcker rule in 2010, named after Alan Greenspan’s predecessor as president of the US central bank, Fed. The Act bans at least proprietary trading of securities for deposit-taking institutes.