An "awkward" situation is uncomfortable because you don't know how to act. In the example at top, the speaker feels awkward because he has already said "goodbye" but now they have to walk with each other. So he doesn't know if he needs to say "goodbye" again, or just stop talking to the coworker.
Some other examples of "awkward" situations are:
- You're on a first date with someone and you run out of things to talk about.
- You're giving a presentation, and the person who introduces you calls you by the wrong name.
- You find out that your teenage son has been having sex, and you have to talk to him about it.
Of course, you don't directly say "This is awkward" in all of those situations. But you would use the word "awkward" to describe the situation when you're telling another person about it.