You do not need a copy printed out, although it is good to have a copy just in case anything happens, but if you have it on your phone you dont need a paper copy.
There are several ways to check in.
1) Home computer. You dont need a printer, you'll still get checked in and can get a reprint of your boarding pass at the airport from either the kiosk or agent. This way you are checked in. If you have a printer, you can print it and go right to the gate without stopping at the counter.
2) Use your confirmation code or bar code (from phone or printout) at a kiosk. You can also use a credit card, atm card, AAA card or any other card with a mag stripe with your name on it (not all kinds of mag stripes work though), your name or a frequent flyer number if its in the record already.
3) If no kiosk, see an agent. This should be the last method of checkin since you might have to wait depending on what else they're doing and if you arent checking bags, there really is no need to speak to anyone at the counter.