The U.S. State Department issues a Travel Warning when they want you to consider very carefully whether you should go to a country at all. Examples of reasons for issuing a Travel Warning might include unstable government, civil war, ongoing intense crime or violence, or frequent terrorist attacks. They want you to know the risks of traveling to these places and to strongly consider not going to them at all. Travel Warnings remain in place until the situation changes; some have been in effect for years.
The U.S. State Department issues a Travel Alert for short-term events they think you should know about when planning travel to a country. Examples of reasons for issuing a Travel Alert might include an election season that is bound to have many strikes, demonstrations, or disturbances; a health alert like an outbreak of H1N1; or evidence of an elevated risk of terrorist attacks. When these short-term events are over, they cancel the Travel Alert. Learn about alerts and warnings for your destination here.
National Terrorism Advisory System
Homeland Security issues an Imminent Threat Alert to warn of a credible, specific, and impending terrorist threat against the United States or an Elevated Threat Alert to warn of a credible terrorist threat against the United States. Find more info on National Terrorism Advisory System alerts here.
Pro tip: Travel warnings are long term while travel alerts are short term.