Liege relationships on Barrayar tended to be complex, with several levels of 'liegeness'.
- "Glad you made it, my lord."
Miles slid into a station chair. "Me too. But please - just 'Mr. Naismith.' No 'my lord.'"
Auson looked puzzled. "The others all call you that."
"Yes, but, um - it's not just a courtesy. It denotes a specific legal relationship. You wouldn't call me 'my husband' even if you heard my wife do so, eh?"
- ―Auson learns what liege relationships are from Miles[src]
For the simplest answer: A liege lord was a person to whom another person had sworn a oath of allegiance.
However, ordinary subjects of Barrayar were born into particular Districts, and they did not seem to ever have had to swear a formal oath to become liege people to their ruling Counts. With Aral Vorkosigan's reform of the rules for moving between Districts and shifting one's oaths, it would appear that not much more was needed for this than the signing of a form.
An armsman's oath was more complicated, and was limited to twenty people per District Count.
There was also a military officer's oath that was taken by anyone who became an officer; the words of this oath are not given in canon, but there existed officers designated by the Emperor for the task of receiving these oaths.