The word games involve finding the longest word you can from 9 random letters. In the actual shows, the contestants take turns in selecting how many vowels and consonants there are.
The number games involve using arithmetic operations to reach a random 3-digit target number using 6 randomly selected numbers.
The very last round is known as the 'conundrum', which involves unravelling a 9-letter anagram
Sounds fairly straightforward - and it is - except that each round lasts only 30 seconds! I think Countdown is so popular because the format allows everyone at home to participate and calculate their score compared to the contestants.
You can play Countdown online, such as HERE or HERE - they all seem to use the same script.
If you want to concentrate on the numbers game, you can try this version HERE. In the actual show, there are times when the target number cannot be reached, but on NRICH they guarantee that every puzzle can be solved.
There are also Countdown mobile apps - just search for them.
Now, I think that these games can be very good at increasing vocabulary and mastery of arithmetic. Note that these are not purely mental challenges; you are expected to have pen and paper available to scribble your attempted answers. I'm obviously not the first to think these games could be appealing to a younger audience, as there was a pilot programme made called Junior Countdown. However, it appears to have been a disaster and was never aired [see Wikipedia]
Well, so much for putting kids on TV! But, you're not on camera, so enjoy the games.