You can count a country or dependency if you deliberately visited it. i.e. Purposely arranged to visit a place and then while you were there achieved something e.g. looking at a museum, exploring a town or doing some business.
Countries and Dependencies must be ones that exist today, not ones that existed in the past. I can't keep a list of all ex-countries as well - I don't have the time. So if you went to East Germany and West Germany, I'm afraid that only counts as one country now.
This works the other way to your benefit, though. For instance, if you once went on a trip all around Czechoslovakia that will now count as two countries - the Czech Republic and the Slovak Republic. You must, of course, have actually visited a bit of Czechoslovakia that is now in the Czech Republic and another bit which is now in Slovakia for this to work.
Q1: I've stopped over in lots of random airports - do these countries count?
A: No, you didn't cross the immigration line.
Q2 from Matthew: Well I've been through Lichstenstein by rail and had to change train there - that must count then.
A: No, you didn't visit these places if you just stopped at a train station and didn't actually see anything.
Q3 from James: I once stopped at Zagreb, Croatia, had four hours to spare while I waited for the next train and spent it exploring Zagreb. Does that count?
A: Well it's a bit borderline but you did plan to do this ahead of time and then actually saw the town while you were there so yes.
Q4 from James: I once stopped in Vienna, Austria for 6 hours while waiting for a train and sat in a nearby park until it came. Does that count?
A: No - you should have made an effort to see the bloody place you lazy oaf.
Q5: Supposing I'm driving to Spain and stop off in the odd French cafe and road-side restaurant en-route. Have I visited France?
A: Fraid not - eating a French stick and having a quick leer at the sexy French girls is enjoyable but does not count as an achievement.
Q6 from Matthew: I made a special effort to visit Hong Kong before it became part of China. I'm going to be mighty dischuffed if that doesn't count.
A: Well it is a bit of a drag but I have to say no. So whether you visited Hong Kong before or after it was handed back to China, it counts as a visit to China.