Is it possible to use WordPress to give users the ability to login to your site and access private content?

Of course it is. You can use any number of plugins do this, but it should be noted that some plugins will work more efficiently than others, some will be easier to install, some will flat out not work, and some will crash your site. So, you have to budget the time to read through each plugin's discussion forum (if it's not current or updated forget about it). I'd look for membership plugins (like S2), and look up several articles Chris Lema wrote on his site (http://chrislema.com) on creating educational and/or membership sites. If you are not an experienced, professional WordPress web developer, however, I wouldn't suggest trying to set it all up on your own.

We use the S2 Member Plugin ourselves and for all of our clients at http://Alphabetacommerce.com.   It is an extremely powerful plugin with tremendous integration and flexibility.