In my effort to provide information on settling down in the United States I already provided details on bank account, credit card, home buying, and drivers license, and this time let me tell you how to get a SSN and an actual card. You see, the Social Security number is a unique identifier that allows you to use it to file taxes, apply for benefits, get a job, etc. Without this card, your life will be impossible. However, the governments here have a vested interest in making sure that you have this number because that is how they will collect taxes from you.
For some immigrants and non-immigrants, the process of getting a Social Security card maybe automatic, but the rest of you may need to make a trip to the nearest Social Security Administration office with your passport, proof of address, and evidence of your legal presence in the United States; complete a form SS5 and submit it with your documents -- that's it; the card should arrive in the mail in a few weeks. Anyone in possession of an immigrant visa or work visa must get a SS# but not all visa holders are allowed to have a number if they are not authorized to work. For instance, tourists cannot apply for a number.
Your SSN is the same for your whole life and you should simply memorize it. There is no reason to carry your card with you in your wallet. If you have ever had a number issued to you and your visa category has changed or you returned to the United States after years/decades in a different category, you can continue to use the same number but you must let the SSA know the change in your status so that your records are up to date. This is particularly important if you get a green card or become a citizen because then your card will no longer say that you need DHS authorization to work. Obviously, if you change your name, you must request a new card and update the information.