Which we use for non-restrictive clause; that we use for restrictive clauses. A non-restrictive clause is used to add non-essential information. Let us say you wish to tell us of your unreliable car. My car, which is ten years old, broke down last Saturday for the eighteenth time. You use 'which' to add ten years old; the information is non-essential as it is not central to the facts of the story. In a sentence using that: I was driving to the store in my car that breaks down all the time. Your central point is that your car breaks down all the time. Notice that which is accompanied by a pair of commas, no such thing occurs when you use that. So: 1) You will hear us talk about the Russian joke again, which I’m fed up with frankly. Yes: You will hear us talk about the Russian joke, which I'm fed up with, again. (That you are fed up of the joke is not the key information, so you can use which: a non-restrictive clause) 2) That’s a very simple game which people play with their friends just for fun. It is a very simple game that people play just for fun. (That it is just for fun and not competitive is vital information, so you need that) 3) I’m now going to read out all the “would you rather” questions which that you’re going to hear in this episode. I'm going to read out all the 'would you rather' question that you're going to hear in this episode (again, that they are going to be heard during this episode is vital so you need to use a restrictive clause). 4) Would you rather… – 
Have accordions for legs? – 
Or have a huge belly button which is 10 inches long and that sways to the beat of popular music? 5) Would you rather – be very very fat (clinically obese)? – or have Tourette’s syndrome? (that’s a mental disorder which means that you randomly shout out very rude words in public and you can’t control it). For both of those you need 'that' as they identify key element in defining the belly, and Tourette's syndrome.

You pose an interesting question Natalya. Here’s what I found in a website (BetterWritingSkills.com) which offers a simple explanation: Use which (surrounded by commas) if a group of words adds information. Use that if it limits the set of things you're talking about. Here are two examples just to make that clear: Classes that are held on Wednesdays are in building 206. Leap years, which have 366 days, contain an extra day in February. In the first sentence, the words "that are held on Wednesdays" are limiting the type of classes that we're talking about. (We're not talking about all the classes, only the ones held on Wednesdays.) We thus use that. In the second sentence, the words "which have 366 days" are adding information. We thus use which surrounded by commas.

Short answer: "That" is used when the relative clause can NOT be omitted without changing the main meaning of the sentence. "Which" is used when the relative clause CAN be omitted without changing the main meaning of the sentence.