Conditions:

1. You must have a sponsor in the United States, or if you are self-sponsoring, you must show financial solvency depending on the process for which you are requesting temporary permission to stay and the amount of time.
2. Apply for:

  Urgent Humanitarian Reasons:

1. Whether or not the circumstances are urgent.
2. The effect of these circumstances on the person's health or well-being.
3. The degree of suffering that may arise if the temporary stay permit is not granted. 

They may be: 

• Petitions for medical reasons.
• Requesting temporary permission to stay for minors.
• Receiving medical treatment in the United States.
• Donating organs to a person in the United States.
• To join a family member in the U.S. for humanitarian reasons.
• To Care for or Provide Support to a Family Member Who Has a Serious or Terminal Illness and Is in the United States.
• To Attend a Funeral or Coordinate Matters Relating to a Deceased Family Member in the United States.
• To Come to the United States to Protect Yourself from Persecution or Personal Injury.
• To Participate in Legal Proceedings in the United States
• Returning to the United States after: Failure to obtain a Travel Document prior to your departure from the United States or Failure to return to the United States before your Travel Document expired.
• Please note that for each case you must have identification documents, proof and demonstrate the urgency, such as cases such as critical medical treatment, to visit or help or assist a family member who is in the final stages of an illness or disease. In addition, it is important to know that for each case the forms to be completed are different.

See here for a guide to the different types of leave granted.

Considerations

1. If the purpose of the temporary stay permit can be achieved within a specific temporary period.
2. If the beneficiary intends to depart the United States when his or her temporary resident permit expires or has the resources to obtain lawful immigration status during the period of authorization of the permit or any period in which he or she anticipates reissuance of the permit (where applicable).
3. If there is evidence of any national security concerns.
4. If there is evidence of criminal history or prior violations of immigration laws.
5. If there is evidence of prior involvement in fraud.
6. Whether the beneficiary's presence may benefit a U.S. citizen or lawful permanent resident or a community in the United States; 7.
7. Whether he or she has sufficient financial assistance while in the United States; 8.
8. Evidence of good moral character.
9. The effect of the beneficiary's presence in a community in the U.S.; and
10. Whether there are any other means, in addition to the temporary permission to stay, that are available to the beneficiary such that the beneficiary can travel and remain in the United States for the stated purpose for the temporary permission to stay described, such as the ability or inability to obtain a visa.

Application

1. File the application for temporary permission to stay: Complete a Form I-131, Application for Travel Document, and include for each temporary permission to stay beneficiary the filing fee or a Form I-912 , Application for Fee Waiver or a written statement for a fee waiver for each application.
2. If there is more than one sponsor, each economic sponsor must submit a Form I-134, Affidavit of Economic Sponsorship .
3. Include a detailed explanation, as well as supporting documentation, as to why temporary permission to stay is being requested for the beneficiary. (See the Submission of Evidence section below for more detailed information, along with the Guidelines on Evidence for Certain Types of Requests for Humanitarian or Significant Public Benefit Permission page).
4. If you are being represented by an attorney or authorized representative, you must include a completed Form G-28, Notice to Appear as Attorney or Authorized Representative for USCIS to contact your attorney or authorized representative.
5. To receive an email and/or text message notifying you when your application has been accepted, complete Form G-1145 , Electronic Notice of Acceptance of Petition/Application and attach it to the first page of your application.
6. Submit your application: Please refer to the Humanitarian Permit Applicants section of our Direct Filing Addresses Form I-131, Application for Travel Document web page for information on where to send your application. Failure to send your application to the correct filing address may delay our processing of the application.
7. Wait for a response: It may take 90 days or more. It is important that you attach as much evidence as possible to prevent USCIS from requesting extra documents, as it may delay the process further.

• If your application for temporary permission to stay is denied and you believe that USCIS decided your case incorrectly, you may file a motion to ask USCIS to reconsider the decision, reopen the record, or both: The requirements for motions are listed in 8 C.F.R. § 103.5. File a motion with Form I-290B, Notice of Appeal or Motion. You must file it within 33 days of the date of the notice of denial and include the correct filing fee or fee waiver.

How to Submit Evidence

• Explain the detailed reasons why you are making the request.
• The amount of time needed for each beneficiary.
• Copy of any prior visa procedures or visa petitions.
• Copy of any evidence of your petition and copy of a government ID of yours indicating your or the beneficiary's citizenship.

How the process works:

1. Files the petition for temporary permission to stay.
2. USCIS reviews the petition for purposes of determining jurisdiction and urgency (prioritization).
3. A USCIS officer makes the decision.
4. A supervisor or designated official reviews the decision.
5. USCIS provides notification of the decision: If the beneficiary appears to be eligible for the permit: We will send a notice of conditional approval or a notice of continued processing of the permit to the petitioner and any authorized representative / If the petition is denied: We will mail a denial letter to the petitioner, beneficiary and any authorized representative.
6. Issuance of Travel Documents and Temporary Permit to Remain in the U.S. (For conditionally approved cases only).
7. Customs and Border Protection (CBP) authorizes permission for temporary stay in the United States (for conditionally approved cases only).

More information here.