There are many schemes that can advertise themselves with certain security notions, usually IND-CPA or IND-CCA2, for example plain ElGamal has IND-CPA security but doesn't provide IND-CCA security.
The most common ones are the "IND-" ones, advertising security against specific classes of attackers.
Now these notion may not be that well-understood by many people so I ask hereby for a canonical answer, that explains what the following security notions mean. A (simple) description of the formal attack scenario is preferred.
Please don't restrict the answer to "you can chose this and if you can break it with this it isn't IND-CCA2". Please at least outline the formal attack (like f.ex. real-or-random). Relations between the "IND-" are explained (f.ex. IND-CCA implies IND-CPA).
The notions in question are: