When you have F-1 status, you are eligible to work on campus and eventually off campus. When employed you are required to have a Social Security Number. However, under current regulations, you must actually have a job offer in order to apply for a Social Security Number. Since you will not initially have a job offer, you will not be able to get a Social Security number immediately.
You do not need a Social Security Number to open a bank account or rent an apartment. However, the lack of an SSN may cause other problems as you arrange for services. Explain to any company asking for a SSN that you are not permitted to get one and inquire if they have an alternative method to provide you the service you want.
If you are offered a job on campus (initially you are eligible only for on-campus jobs), the Human Resources office (HR) in Carpenter Hall will give you a letter verifying your job offer. If you are offered a job in the cafeteria or bookstore, Aramark (food service) or Follett (bookstore) will give you a similar letter.
Bring your job offer letter to the International Office. There you will receive another letter verifying your F-1 status. With these letters, your I-20 and passport with I-94, you'll be able to go to the Social Security office and apply for a Social Security Card. The card will be sent to you in about 2 weeks.
Employers are not required to wait until you have the card before employing you. However, some employers do choose to wait.