Living vicariously through another is to enjoy their life and experiences as if they were your own. This often takes place when one sees someone living a life they wish they had.
In some cases, people may push things that they would like to do on other people in order to live vicariously through them. For example, Tyler Tervooren from the Riskology website is a risk taker who often receives emails from people suggesting risks he should take. He believes that, in some cases, these are people who wish to live vicariously through him.
Parents sometimes try to live vicariously through their children, and they may push their children to do things that they wish they had accomplished.