The refund process
1) You cancel the trip prior to day of departure. You will receive a notice of cancellation and a FFC

2) You have 1 year from the original purchase date to reschedule travel (the new trip has to start within that one year period). The new travel must be in the same name as the original ticket. Keep the record locator (PNR) and e-ticket number to reference when rescheduling. (This will be a FFC)

3) You will be able to apply the full purchase price of the original trip toward the new trip.

4) If the new trip is less expensive, you will receive an FFC good for future travel. .

5) If requesting a refund, you need to applied to UA for a medical waiver (E-Ticket refund). When applying for the refund you will need to provide documentation (there is a method for uploading documents)In the case of relative or traveling companion's death: Please provide a death certificate.

In the cases where there is a change fee, pay the change fee and after the flight file for a change refund
If you are sick: Please provide a doctor’s note advising you not to fly.
If you have an immediate family who is sick: Please provide the name, address and telephone number of the hospital where they are admitted and the name of their doctor.
If you have jury duty: Please provide a copy of your jury duty notice.

Note: If the medical certification forbids travel in the one year period, a full refund to original method of payment (versus e-voucher) is possible.
There is no longer a fee, United Airlines drops $50 fee for hardship refunds Nov 2015

From UA's Contract of Carriage {13 Dec 2013}
Archive thread: Consolidated "Refunds/Cancellations Due to Illness/Death" Thread {Archive}