An immigration waiver is a request to be forgiven for an immigration law violation.
Not all violations are waiver-eligible. There are though, waivers available for both visitor's visas and immigrant visas as well. Some waivers are available to people currently in the U.S. while others may apply to people outside America who want permission to enter.
Usually the need to seek a waiver comes up because a person was (1) overstayed in the U.S. past your visa expiration, (2) you were removed from the U.S., or (3) charged or convicted in a criminal matter.
The most common waivers are:
I-601 Waivers: they seek to waive removal for spouses, fiancés and children of U.S. citizens or green card holders. “Extreme hardship” evidence must be presented.
I-601A Waivers: they seek to allow certain relative of U.S. citizens and green card holders to apply, in advance for a waiver of inadmissibility. If granted it allows the applicant to leave the U.S. and return to their home country to appear at a U.S. Embassy for an immigrant visa interview. The program's advantage is it allows the applicant the chance to return home with some reasonable assurance of getting a waiver and the time out of the country can be just a matter of days. Again, the key element is a showing of extreme hardship to your family.
I-212 Waivers: they seek to allow non-U.S. citizens to apply for early readmission into the U.S. if they were previously removed and they want to return before their statutory exclusionary period expires (they have a desire to return before their 3 or 10-year ban is fulfilled)