Definition of investment bank
Also called merchant bank (particularly in the UK), this is a bank with a wide range of specialised services for companies and large investors, including underwriting and advising on securities issues and other forms of capital raising, mergers and acquisitions, trading on capital markets, research and private equity investments etc. Also, an investment bank trades and invests on its own account. 
A financial institution that deals mainly with corporate customers and specialises in securities markets activities including underwriting, trading, asset management, advisory activities and corporate restructuring such as mergers and acquisitions.
What are the main differences between commercial and investment banking?
Commercial banking relates to deposit-taking and lending whereas investment banking is predominantly a securities business. Traditionally commercial banking was viewed as relatively less risky than the more volatile investment banking business. Commercial bank performance was very much linked to economic growth and credit demand whereas investment bank performance is strongly influenced by stock market performance.
In many regions/ jurisdictions commercial banking and investment banking have been legally separated. For instance, in the US, the 1933 Glass-Steagall Act prohibited commercial banks from carrying out investment banking and insurance business. This legislation was put in place by the US authorities in response to the 1929 Wall Street Crash where excessive bank securities speculation was blamed for the crash and the Great Depression that followed.
Similar legislation was put in place in Japan in 1948. This legislation was repealed in 1999 in both countries. Banks in the EU have been allowed to offer commercial and investment banking services without restriction since 1992.
Note that many commercial banks do investment banking business although the latter is not considered the main business area.
Since the 2007-8 banking crisis, few stand-alone investment banks remain. Goldman Sachs and Merrill Lynch, merged with Bank of America (to become Bank of America Merrill Lynch), converted to bank holding companies in 2008 mainly to increase protection and to enable them to extend their deposit taking business.
Goldman Sachs and Morgan Stanley are the last remaining major Wall Street firms that operate as independent investment banks. Others are now part of major commercial banks, such as Merrill Lynch, and other smaller operators such as Lehman Brothers went bust in the 2008 financial crisis.