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.
Why one should use the idioms, and what do the idioms accomplish?
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.