Full employment can mean various things. The most literal interpretation would be a situation where everyone has a job, besides those who are not part of the labor force. Even this definition is somewhat tricky, since many people will enter and leave the labor force depending on the strength of the economy. Whether teenagers or retirees should be considered part of "full employment" is an open question. Also, there will always be some frictional unemployment, which is defined as people who are unemployed while looking for jobs. So even this definition of "full employment" includes some unemployed people.
Generally, however, "full employment" refers to a more specific concept in economics. An economy can be considered to be in "full employment" when as many people are as employed as possible, without contributing to the rate of inflation. When too many people are employed, the price of labor, also known as wages, go up, because businesses are competing for a small pool of employees. This either leads to an inflationary spiral, or else rising costs force businesses to stop hiring, and the unemployment rate goes back up to a point where it doesn't lead to inflation. There is much very detailed work in economics that explains the actual mathematical relationship between unemployment and inflation, but the basic relationship between too high employment and inflation is a matter of consensus in mainstream economics. The exact amount of unemployment that is considered healthy is usually around 5%. I have been wondering lately about whether the current high rate of unemployment in the United States, around 10%, is the result of a recession, or whether the rate of full employment has just dropped.
Which brings up another big issue around the full employment number: besides its economic function, employment also has a very important cultural and social role. To wit: to an economist, 5% of people not having a job is a good, normal thing. But when you are that person, trying to explain to those around you why you are sleeping until noon, just to wake up and eat Doritoes and play video games all day, you can't really pull out a bunch of charts showing how you are helping to keep inflation down. In other words, cultural values towards employment are not totally in line with how economic science views employment. I think that there is a great reluctance amongst politicians to deal with this problem, just because it is socially unacceptable to admit that unemployment is a normal feature of the economy.