Let's take the symmetric group S3, which multiplitation table is
I understand that its irreducible representation is the set of 2x2 matrices
But S3 has also an unfaithful representation where {e,d,f} goes to 1 and {a,b,c} goes to -1. Is this representation also irreducible? I think it isn't but I can't really argue why. Does Irreducible representations have to be faithful?
Irreducible representations definitely don't have to be faithful. Indeed, irreducible representations are representations that are as "small" as possible, which makes it harder for them to be faithful than general representations. For instance, if $G$ is any group, there is a trivial $1$-dimensional representation of $G$ that sends every element of $G$ to $1$. This is as far from faithful as possible, but is always irreducible.
More generally, any $1$-dimensional representation $V$ is irreducible: the only vector subspaces of $V$ are $0$ and $V$ itself, and so in particular there cannot be any subrepresentations of $V$ besides $0$ and $V$. In particular, this applies to the $1$-dimensional representation of $S_3$ you are asking about.