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. Some employers may choose to wait until you have the card. However, you will generally not get paid until you have the card.