Like Facebook, the system allows users to add friends and share updates, photos, videos and links. Unlike Facebook, Google+ also offers Hangouts (group video chat), Circles (a better way of grouping your contacts) and Sparks (a not very well integrated way of subscribing to topics).
Google+ was originally opened up to a select bunch of übergeeks, who in turn started to invite friends onto the service. Then Google closed the invites system. So now it seems you just have to wait....
...Or do you?
I tested this method found via a friend on Google+ and it worked. Firstly, you need to know someone who's already on Google+. Then, get them to create a new Circle - it doesn't matter what it's called. Next, have your friend add you to the Circle, which you can do by inserting an email address and then a name. Finally, send an update to that Circle, checking the 'Also email x people not yet using Google+' checkbox before sending the update.
If everything goes according to plan, you should get an email from Google+ that includes a link to view the new update. Clicking the link takes you to a page that allows you to join the service immediately.
Google is currently throttling the number of people who can join via this method. Therefore, it may not work all the time. But keep trying and you should get lucky soon.
If you don't know anyone who's already on Google+, leave a comment below and I'll do my best to invite you when Google relaxes invitation controls. I can't guarantee that everyone will get an invitation but I'll do what I can.