I heard that the Boston Marathon bomber, Tsarnaev, can face the death penalty. So this means that Massachusetts has the death penalty? What about other states?

This is a really good question.  The answers are:

1. Yes, Tsarnaev could receive the death penalty.
2. No, Massachusetts does not have a death penalty.
3. Yes, other states do impose the death penalty.

The US has both federal and state court systems.

Remember, because of federalism , the United States has both a federal court system and also state court systems.   Tsarnaev is currently on trial in a federal court, not state court.  The federal court is located in Massachusetts but the defendant is  facing federal charges, not state charges.  For some of his crimes, the federal court could sentence Tsarnaev to death.   

Some, but not all states, impose the death penalty for certain crimes.  Massachusetts abolished the death penalty but that will not help Tsarnaev in federal court.  

Tsarnaev could be tried on state charges, too.

After his federal trial is over, Tsarnaev could be tried again on state charges in a Massachusetts state court.  That court could not punish Tsarnaev with the death penalty.

Below is a video discussing federalism: