Deportation is obviously a very touchy subject, no matter who you are. On the one hand, people who are here in breach of the rules and laws obviously are putting themselves in a position of legal punishment. But at the same time, they're people just like anyone else, and you can't help but feel some sympathy for their situation. Maybe they are fleeing a truly terrible situation in their home country.
People who are up for deportation or removal obviously fear being caught. For the most part, they are good people who have no desire to harm anyone or any country. They are just trying to live the best life they possibly can. Still, they can be deported and this can throw their life, or the lives of their family members, into turmoil.
This is where certain waivers can come into play to help someone, or their family members, stave off deportation or some of the consequences associated with deportation. "Extreme hardship" and the "601 waiver" are examples of reprieve that an individual who is in the country illegally can obtain to help their situation.
Now, it's important to remember that there is no guarantee that your waivers or appeals will be accepted. However, they could be very helpful. For example, the 601 waiver can eliminate the 10-year banishment period that is applied to someone who was in the country illegally and discovered. These people are then removed from the U.S. and given a 10-year ban. The 601 waiver eliminates that 10-year ban.
Extreme hardship works in tandem with the 601 waiver. It is a threshold that must be proven by the individual to show they deserve a 601 waiver. There are many criteria that can prove extreme hardship, but simply having a family member in the U.S. or a family member who is a U.S. citizen does not guarantee extreme hardship. Significant medical issues or physical conditions can be enough to qualify someone for extreme hardship.
Source: FindLaw, "Extreme Hardship and the 601 Waiver: Can I Avoid Deportation?," Accessed April 7, 2015