The Constitution of the United States of America does not mention local governments.  Local governments are created by and regulated by the states.  This means that to speak about cities or other forms of local government in the United States is to speak about fifty different legal and political situations.

The states outline the powers of municipal governments in charters.  There may be:

• special or specific charters;
• general or classified charters (in which the rules may apply to a class of cities,
  often grouped by population size); or
• home rule charters.

The state municipal leagues can provide information about the form of charter provided the state constitutions.

Generally, cities in a "Home Rule" state have greater autonomy, while cities in a "Dillon's Rule" state are more restricted in their powers.  For more information, see Dillon's Rule.

Home rule is a delegation of power from the state to its sub-units of governments (including counties, municipalities, towns or townships, or villages).  That power is limited to specific fields, and subject to constant judicial interpretation.  Home rule creates local autonomy and limits the degree of state interference in local affairs. 

The powers and limits of home rule authority for local governments are defined state-by-state.  There is no one national standard that defines the powers engendered in home rule status.  State provisions for home rule by its local government entities can be defined by the state's constitution and/or statutes enacted by its legislature.  Not all cities make use of the discretionary powers of home rule that are provided by their charter.   Functional powers are the most frequently used and expanded.
 
There are four primary areas in which "home rule" powers are exercised by governments:

• Structural -- power to choose the form of government, charter and enact charter revisions;
• Functional -- power to exercise powers of local self government; sometimes qualified as "broad functional" or
  "limited functional" home rule with varying degrees of autonomy;
• Fiscal -- authority to determine revenue sources, set tax rates, borrow funds, and other related activities; and
• Personnel -- authority to set employment rules and conditions ranging from remuneration to collective bargaining.

Sources:
The Urban Politics Dictionary. Santa Barbara, CA: ABC-CLIO, 1990.

United States Advisory Commission on Intergovernmental Relations. Measuring Local Discretionary Authority. M-131. Washington, DC: U. S. Advisory Commission on Intergovernmental Relations, 1981. (not available on web)

United States Advisory Commission on Intergovernmental Relations. State Laws Governing Local Government Structure and Administration. M-186.  Washington, DC: U. S. Advisory Commission on Intergovernmental Relations,1993.