Someone asked this on another forum the other day, my reply was...
It depends on the organisation, for instance my last place:
1st line was the helpdesk, however they fixed a lot of calls there and were not just merely call loggers.
2nd line was for things that needed desk/site visits or were too time consuming to do on the helpdesk. Not an indication of how difficult something was, purely time based really, for most things anyway.
3rd line, well wasn't really anything in house I'd have called third line. This was when we'd call up the manufacturer or whoever we had a support contract with.
Historically the same big team covered 1st and 2nd line duties as well as all the project work, with I guess some people in the team acting as 3rd line when the 2nd line person couldn't fix it but someone else in the team had those skills.
Now since I've left they've split the teams up now into smaller groups so the 1st line people are 1 small team and all they do is 1st line, same for 2nd line etc. However even then the first line is not merely a log it and shove it job, they actually know what they are doing and are reasonably technical.
Where I am now the desktop guys cover 1st and 2nd line, then if they can't do it it either comes to one of us server guys to help or whoever we have a support contract with.
So I guess I'm 3rd line by our definitions, but tbh we do what we want to most of the time and help out when others get stuck.
This week our desktop guy is off and the new boy doesn't start until Monday so we're having ot do everything, joy. Although the guy who moved to us from the desktop team has been stuck with doing most of it