One of the things that confuse me most when I study group theory and quantum field theory is that I constantly run into the situations where $\psi C \Gamma_M\chi$ are treated like vectors, $\psi C\Gamma_M\Gamma_N \Gamma_N \chi$ are treated like tensors, where $\psi$ and $\chi$ are spinors in the SO group's spinor representation.
But I never have been formally introduced that tensors and vectors in this way. I currently feels that maybe many traditional vectors (like what we learned since high school) can be represented in this way by the contraction with $\gamma$ matrices?
So back to the topic, what does it mean when some author says that "they are like vectors and tensors"? Would these objects completely satisfy the definition of tensors? How do we manipulate the transformations of them; would they transform like tensors?
P.S.: I'm a physics student, so I would really appreciate it if your answer not heavily relies on math terms. Thank you in advance.