I lost my house to foreclosure and need more time before moving. They want me to sign an Agreement for Judgment
What is this and what will it mean for my legal rights?

It means that the court judgment will be whatever the agreement says. If it provides that, for example, you will have to be out by a certain date, that is what the judgment (final court order) will say. If you don't comply with it you will be in contempt of court.

The agreement may also give them a deficiency judgment (what you owe less whatever the property brings at the judicial sale). The can obtain a writ of execution and seize your non-exempt property to satisfy the judgment.