Let $A,B\subseteq \Sigma^*$ be languages.
Many-to-one: A (computable) function $f:\Sigma^*\to \Sigma^*$ such that $\forall x\in \Sigma^*$, $x\in A\iff f(x)\in B$.
The names "Mapping reduction" and "Karp reductions", to my knowledge, refer to "Many to one".
The "Many to one" means that $f$ may not be injective.
Turing reduction: we say that $A\le_T B$ if, given an oracle to the language $B$, we can use it to solve $A$. The word "solve" here should be in the context of a specific complexity/computability class.
Turing reductions are weaker than many-to-one reductions. The latter can be viewed as Turing reductions where we are only allowed to call the oracle once - at the very end of the run.
polynomial time many-to-one reductions - simply adding a constraint that the reduction $f$ is computable in polynomial time.
polynomial time Turing reduction (= Cook reduction) - add the constraint that the oracle machine runs in polynomial time, counting each oracle call as $O(1)$.