Something or someone that is embarrassing, uncomfortable or difficult to deal with is awkward.
EX: I used to have a very awkward job at the bank. I had to call people who were late on their mortgage payments and explain to them that the bank might take their house if they didn't make a payment soon. I really hated it.
EX: Elaine is a very awkward dancer. She takes up a lot of space, it's impossible to ignore her, and yet she really doesn't keep time with the music. People don't know what to think.
A person who often makes people uncomfortable in social situations and says inappropriate things is socially awkward. When two or more people stop talking for a long enough moment that they begin to feel uncomfortable, you can call this an awkward silence. And when you ask someone to do something that is uncomfortable or embarrassing for them, you put them in an awkward position.
EX: Our date did not go very well. We seemed to have a lot of common interests, but we didn't seem to talk about them for very long. There were a lot of awkward silences and I really wasn't sure if he was interested in talking to me at all.
EX: Rita doesn't really know her bosses very well as she's just started in the company. So asking her to recommend you to the human resources manager might really be putting her in an awkward position.