If the local office of the Illinois Department of Employment Security (IDES) denies you unemployment benefits, you can appeal this decision.
After you've applied for unemployment benefits, you will get a letter from IDES which will state whether you have been approved or denied. You must appeal within the legal time limit which is usually 30 days, but may be shorter. Make sure you do not miss the time you are given to appeal, because late appeals will not be considered.
Keep in mind that even if you are initially found to be entitled to benefits, your former employer has a right to appeal the decision within 30 days. So you should wait for 30 days until you rely on this decision to be final. You will still receive benefits in this 30 day period.
The letter you receive denying you unemployment benefits explains how to appeal the denial. You can follow the step by step process here. Do not wait until the last week to submit your appeal, so that if your letter is not delivered, you will have time to send or hand-deliver another one.
If there is an appeal, you will have a hearing with a Referee.