What kinds of problems have caused Americans to amend their Constitution?

Congress and the states have not sought to solve every problem faced by the United States through changing the Constitution. In general, amendments to the Constitution are not usually made when simply changing federal law would be sufficient. The Prohibition Amendment (Amendment 18) could have simply been passed as a combination of state and federal law, but was put in the Constitution largely in order to make it difficult to repeal. Nevertheless, the 18th Amendment was repealed by the 21st Amendment. 

Most of the amendments to the Constitution can be categorized as seeking to limit the power of government, expand the power of government, change election procedure or duration of office, or expand the electorate. 

Limiting the Power of Government
Amendment  Purpose
I Limit power of Congress to establish religion, abridge speech, press, assembly, or right to petition
II Limit power of Congress or Executive to infringe on right to bear arms
III Limit power of Executive to arbitrarily use private homes to house militia
IV Limit power of Executive to engage in search and seizure of evidence without court approval of probable cause
V Limit power of Courts to hold serious trials without grand jury, to compel testimony against oneself, and to try a person twice for the same crime. Limit power of Government to take property without just compensation
VI Limit power of Courts to hold trials that are not speedy, do not allow suspects to know charges against them, or do not allow the accused to confront witnesses against them
VII Limit power of Courts to hear civil trials without a jury
VIII Limit power of Courts to impose excessive bail or cruel and unusual punishment
IX Limit power of Government to disparage non-enumerated rights retained by the people
X Limit power of National Government to use powers not specifically delegated to it in the Constitution
XI Limit power of National Courts to hear lawsuits against state governments by individuals
XIV Limit power of National and  State Governments to deny citizenship rights to any person born or naturalized in the United States, deprive any citizen equal protection of law, or deprive any citizen of life, liberty, or property without due process
XVII  Limit power of Congress to raise its salary

Expanding the Power of Government
Amendment Purpose
XVI Allow the National Government to directly tax incomes 

Changing Election Procedure or Duration of Terms of Federal Offices
Amendment Purpose
XII  Provide separate ballot for vice president in the electoral college 
XVII  Provide direct election of senators 
XX  Eliminate long "lame duck" session of Congress 
XXII  Limit presidential term 
XXV  Provide presidential succession in case of disability 

Expanding the Electorate
Amendment  Purpose 
XV  Extend voting rights to all races 
XIX  Extend voting rights to women 
XXIII  Extend voting rights to residents of the District of Columbia 
XXIV  Extend voting rights to all classes by prohibition of poll taxes 
XXVI  Extend voting rights citizens aged 18 and older 

Legislating
Amendment  Purpose 
XIII  End slavery 
XVIII  Prohibition of the manufacture, sale, or transportation of intoxicating liquors 
XXI  Nullify prohibition amendment