How are viewing figures gathered?

What you are referring to is commonly known as TV ratings.

So how do television companies judge how many people have viewed their shows?

They usually rely on two main techniques, applied to a representative minority of viewers who get payed in helping collect the TV ratings:

1. Viewer diaries - The viewer writes down what he watches when. This information is then collected and analyzed regularly. This method is cheap and easy, but of course not very accurate.
2. Set meters and people meters - These are set top boxes connected to the TV that gather viewing habits on a very detailed level (per-minute basis). Those boxes are activated by the viewer when he starts to watch a TV show, people meters even track who exactly in a household is currently watching. The results then get transmitted to the ratings authority.

In the US this is called Nielsen ratings by the Nielsen Company, the UK has the Broadcasters' Audience Research Board and in Germany the ratings are currently surveyed by the "Arbeitsgemeinschaft Fernsehforschung".

The overall ratings are then extrapolated from the representative small data set.

Other techniques that also get applied include

• Surveys - People are asked what they watched.
• Internet viewership - The click numbers of online services are getting more and more important nowadays and are getting more attention lately.

I have personally received Nielsen Ratings, even as late as a few years ago. Their web site says:

Electronic and proprietary metering technology is at the heart of Nielsen audience measurement. In addition to capturing what channels viewers are watching on each television set in the home, our meters can identify who is watching and when, including “time-shifted” viewing—the watching of recorded programming up to seven days after an original broadcast.

Chosen at random through proven methodology, Nielsen’s U.S. TV families represent a cross-section of representative homes throughout the country. We measure viewing using our national and local people meters, which capture information about what’s being viewed and when, and in the major U.S. markets, specifically who and how many people are watching. We also have TV set meters in many local markets, and we collect more than two million paper diaries from audiences across the country each year during “sweeps”—specific periods during the months of February, May, July and November. To measure video content viewed on mobile devices, we have expanded our panels to incorporate census-style data from third parties in order to capture the breadth and depth of consumer usage.

Nielsen delivers a constant, real-time stream of information, revealing tuning behavior during programs and commercials. We can tell clients which commercials are being watched, as well as which ones have the strongest engagement and impact. We analyze which position in the program or commercial block is most effective for a specific brand and which markets will create the best return on investment for brands.

I know Nielsen is just one ratings company, but would bet the other ones are pretty much the same.

In the U.S., Nielsen would send out set-top boxes that would physically attach to your cable box. Those boxes would sending viewing information back to Nielsen, who would compile it and sell the data to TV stations so they could use it to charge higher advertising rates during programs with higher viewing rates.