There are some things having a credit card makes much easier -- reserving a hotel room or renting a car, for example, or ordering stuff on line. The effect on apartment-rental options should be minimal, but in some markets having a card (or more accurately having a credit rating) may make a difference.
I didn't get a credit card until I was 22 or so. (I didn't get a driver's license until then either.) At the time, that wasn't a problem at all. I suspect that, really, it still isn't -- if you don't have a specific need for a credit card, you really may not need one. It does help provide a bit of emergency safety net, which may be why your mother is suggesting uou get one -- but it also may present a temptation to buy things faster than you can really afford them, which is a nasty trap to get into. The interest rate on credit cards is moderately obscene.
If you can get a no-yearly-fee card, and if you can trust yourself to always pay off any charges immediately short of a true emergency, having the card will do no harm and may be helpful. If you have to pay for it, or if you expect to ever be tempted to keep a balance on it for more than 30 days, think twice.