Join Types: Whats the difference & how do I know which one to use?

Summary

What are the different join types and how do you determine which join type to use when building your report?

Content

So I fully admit I'm just learning all this BI report building and I cant wait to go to OHUG this year so I can get some face-time with all you experts and learn more about building and editing reports, but I'm struggling a bit right now. Our team is trying to wrap our collective heads around joins. 

We built a report, ran it and it didn't work, went back, changed the type of join, ran it, same result, changed the join to a different join, ran it again and then it finally worked. What we'd like to know is: