If you'd like an invite send an email and we'll see what we can do. Google seems to be hitting limits still once in a while. (I'm sure we won't be able invite everyone)
Signup for Google+:
You will need a google attached account, either gmail or another email you have signed up with.
Google has a nasty habit of wanting to verify you via text message to your cell phone or calling you at some number (this is not specific to Google+ but in general for all Google accounts).
You can add email addresses or people that are on Google+ to your circles. If they are not on Google+ they will get an email when you share something with them or a circle they are in.
Your circles are like groups of people with you can communicate/share with.
Note unlike facebook when you add someone to your circle, they don't have to add you to theirs. So sharing can end up being one way (depending on the permissions you have assigned to the circle they are in.)
Your streams (Nearby (mobile), Stream and Incoming):
In your Stream you can see posts from you and people that have added you to their circles and shared things with those circles. Nearby on android app are public posts from people nearby your location if it is known. Incoming is for posts shared with you but where the person is not in one of your circles.
(A little confusing isn't it all).
Facebook made sharing somewhat symmetrically requiring both friends to accept before full sharing could happen. That is not the way it works on Google+, you can share with someone that does not share back. I suspect folks used to Facebook (read everyone), may have some trouble keeping up with this.
Are like chat groups where you can also do video. (I ahven't tried this yet).
You can post/share with entire circles, all circles, all your circles and your friends circles or variations of those.
Many items defaulted to be accessed by anyone on the internet. Some visibility settings won't let you change it to private even though the option is there (perhaps a bug?) Overall there seem to be significantly less options to control what you want people in your circles to be able to do as compared to facebook. They do have a nice layout though putting the setting right next to the profile item on your profile, which makes it much easier to see how things are configured and change them.
What seems missing compared to facebook:
Applications are missing. If you don't use any or are annoyed anyway by farmville this may not be a big deal. Pages are also missing, which means its hard to link interests, etc to relevant content. No fan pages or user pages for public figures that want to maintain a separate page. There are no groups yet so there is no way to create discussion forums for groups of people. Notes are missing, there is no concept like this yet. Bling, etc also is not available. Finally people are missing, while there may be 10million people, unless you invite all of your friends.
There is no writing to someones wall. (This is a big one. In some ways it changes the nature of sharing for some people that are used to having this enabled).
There is also no integrated message system like email for private messages between G+ users. (Yes there is Gmail but its not really integrated with the notification system, etc.)
What Google+ has that facebook doesn't (by default):
One way sharing
Viewing public posts of nearby users (mobile)
More export of data options
Video chat (Huddles)
Google+ seems like a basic implementation reboot of facebook. For those that hoped privacy would be better there doesn't to be any sign that it will be with many items being defaulted to be visible to anyone on the web. The concept of one way sharing with circles is liable to throw many facebookers for a loop when they add a friend and they don't get added back (thus one way sharing could happen accidentally or purposely.) The mobile platform was about as well featured as the facebook equivalent on android up until recently when facebook seemed to race to add pages to their app plus some additional niceties. Google+ just hit 10million users but it still seems really quiet there and I think lots of folks will likely not want to move to google's platform until it is a better match to facebook and more of their friends are there. Google+ still needs a lot more features and users if it wants to even get close to Facebook. I do applaud the attempt and look forward to monitoring the improvements they make.