Accepting a request to join
If you're a president or vice president, you may encounter one of these windows:
Which should be pretty self-explanatory.
Inviting people to your club
I did this backwards, for right now please just focus on the labeling of "invite to a group" (which really should be "invite to a club >_<). Once you click it this window will pop up:
Select the badge of the club you want to invite the person to, and click invite. If they join, you'll get a notification near your Leave (おでかり) button at the bottom of your screen.
And now what you've all probably been waiting for ...
Joining a club
(Repeating this image.)
When you click on someone and open their pop-up profile, you can see the badges of the clubs they're part of. If you click on one of these badges you will get the left window. Clicking "join club" will give you this window:
Where you can enter a message for the admin of the group, but I believe it's optional.
If you're accepted, you should get a notification near your Leave (おでかり) button at the bottom of your screen. (I believe.)
Sorry is this is a bit messy. If you have any further questions, feel free to ask. I'll try my best to help, and if I don't know the answer hopefully someone else will :D