When a person commits a capital crime, a mitigating factor is something that might help the person avoid getting the death penalty.  ("To mitigate" means "to decrease."  A "factor" is something that causes something else.  So, in law, a mitigating factor is something that can cause punishment to decrease.)

Mitigating factors do not automatically result in decreased punishment.  Judges and juries also think about aggravating factors  things that are likely to cause a more severe punishment. 

Also, mitigating factors are not an excuse for committing a crime.  They may help explain what caused a person to commit a crime, but they do not mean the person did nothing wrong.

Importance
In the United States, mitigating factors are very important in death penalty cases.  Mitigating factors can help prosecutors decide whether to ask for the death penalty.  Also, the United States Supreme Court has ruled several times that judges and juries must think about mitigating factors before deciding on a sentence.

Supreme Court rulings

Since 1972, the Supreme Court has tried to make capital punishment in the United States fairer and more consistent.  The Court has ruled that state laws cannot say a person must get the death penalty if they commit a certain crime, no matter what.  To be fair, the Court ruled, each judge or jury has to think about whether an individual defendant deserves the death penalty.  As Supreme Court Justice Harry Blackmun wrote in a 1994 case: 

As part of this process, the judge or jury has to think about mitigating factors that might make a less severe punishment fit better.

In Lockett v. Ohio, , the Court ruled that state laws may not limit what mitigating factors judges and juries can consider.  The Court added that mitigating factors should include:
 Things about the defendant's character (what they are like, and how they act)
 What the defendant has done in the past
 Anything about the crime that the defendant brings up as an argument for why they should not get the death penalty

In other words, as legal scholar Jeffrey Kirchmeier explains:

Each state has its own death penalty law.  Each law gives a list of capital crimes in the state; aggravating factors that can make person more likely to get the death penalty; and specific mitigating factors.  However, the mitigating factors listed in each state law are only examples of possible mitigating factors.  Even if a state does not list a certain thing as a mitigating factor, a defendant can still bring up that factor in court.

Federal law
The United States' federal courts can try people for federal crimes and can sentence them to death.  Federal law includes a death penalty law, 18 U.S. Code § 3592.  It says that when deciding whether or not to give a defendant the death penalty, the judge or jury must "consider any factor, including these:
 The defendant was not able to understand how wrong their crime was (they had "impaired capacity")
 The defendant was forced or severely pressured to commit the crime
 Somebody else committed most of the crime
 Another defendant who is just as guilty will not be given the death penalty
 The defendant never committed a crime before
 The defendant committed the crime while they were badly mentally ill or having severe emotional problems
 The victim agreed to the criminal behavior that caused their death
 Any other things about the defendant's past, behavior, crime, or anything else that might be a mitigating factor

State laws
While each state law is a little different, most of them list the same mitigating factors as the federal law.  Examples of other mitigating factors in some of the states' laws include:
 Age (the defendant is old enough to be given the death penalty, but is still young)
 Intoxication with drugs or alcohol
 Cooperation with law enforcement officers and prosecutors
 The defendant was severely abused as a child, or was abused by the victim

Oklahoma's sentencing law, Code Section OUJI-CR 4-78, just says that mitigating factors are:

References

Capital punishment
Courts (law)
United States law