What exactly is a signature campaign?
It's a campaign a company or an individual wants to launch, which pays forum users to advertise their service to their signature. If you have signatures enabled (which is the default option once you register), you can see that I have a signature under this post, which points to "ChipMixer". It indicates that I'm part of that signature campaign.
and what does a manager do?
The manager takes care of the campaign. He ought to cautiously select the members of the campaign (according to their quality content, post count, merit count etc), check their posts weekly, update the spreadsheet accordingly-- generally, he's in charge of the service's in-forum advertising.
Here's a list of signature campaigns that are run currently:
https://bitcointalk.org/index.php?topic=615953.0