M2 is a narrow measure of money supply. The UK definition is equivalent to the retail component of M4. This is:
Banknotes and coins in circulation plus retail deposits with banks and building societies.
Retail deposits are deposits made at a banks' and building societies' advertised rates. It excludes very large deposits for which banks are willing to negotiate interest rates.
Notes and coins in circulation excludes those held by non-residents, financial institutions and the public sector.
Definitions used in other countries are very similar. It is usually broadly equivalent to M1 plus savings accounts.
See M1, M0 and money multiplier for comments on government control the money supply.