What C++ idioms should C++ programmers know?
By C++ idioms, I mean design patterns or way of doing certain things that are only applicable for C++ or more applicable for C++ than most other languages.
Please explain why one should use the idioms and what the idioms do.
By far the single most important "pattern" to learn and know that's (nearly) unique to C++ is RAII (Resource Acquisition Is Initialization).
Edit: (To answer extra question edited into the question). You use RAII primarily to (semi-)automate resource management. The most obvious use is freeing resources owned by objects when the owning objects go out of scope, such as freeing memory or closing files.
Template metaprogramming. It's great because it's basically compile-time duck typing, so you get most of the flexibility of duck typing with the speed of static typing.
If you want to get the most out of the STL then iterators and functors/function objects are essential idioms. The use of iterators also implicitly relies on the 'half-open range' idiom too.
Here is one list. If I had to pick a couple I might go with the Curiously Recurring Template Pattern or Virtual Contstructors.
RAII, COW, pimpl, law of demeter (not sure if can be classified as idiom), type traits and policies. (COW and law of demeter are not limited to C++ though)
I prohibit default copy constructors and assignment operators. I actually go beyond that, but those are the most common. I think life would be easier if they were not implicit.