I would like to invite someone external to my organization to be part of a Skype for Business meeting which I am setting up in Outlook. I sent him a connection request using the Skype ID he gave me, but it doesn't seem to be working, so he asked me for my Skype ID so we can connect that way. Where can I find out what my ID is in Skype for Business?
In Skype for Business, your Sign In Address functions as an ID.
However, knowing that probably won't help, since it sounds like you have a different issue. By default, Skype for Business networks cannot communicate with regular Skype (personal) accounts.
This article from Microsoft describes how to enable communication between your Skype for Business and regular Skype accounts.
However, it's a bit more complicated. At least according to this other article from Microsoft, it's still not possible for Skype users to search for Skype for Business users at all.
"While you can search for and find Skype users, they can not search for and find Skype for Business users."
If that seems surprising and frustrating, bear in mind that Skype and Skype for Business are actually two different services. They have very little to do with each other, and are not particularly integrated.