A vector that transforms like a four-vector What is the criterion or what's the meaning when saying that a vector transforms like a four-vector?
 A: When we say that an object transforms like a four-vector, we mean that the object consists of a set of four quantities $A^\mu$ where $\mu=0\ldots 3$, and that if one transforms to a new coordinate system via a Lorentz transformation, then the transformed quantities are given by
$$ A'^\mu = \Lambda ^\mu _{\ \ \ \nu} A^{\nu}$$
where $\Lambda^\mu_{\ \ \nu}$ are the components of the Lorentz transformation matrix.
An example of a situation in which one might say that a set of objects transforms like a four-vector (rather than that the set of four objects is a four vector) is the Dirac gamma matrices $\gamma^\mu$.  In this case, each $\gamma$ is a 4x4 matrix in its own right, so the collection of matrices is obviously not a vector in the usual sense - nonetheless, it can be shown that under Lorentz transformations,
$$\gamma'^\mu = \Lambda^\mu_{\ \ \nu} \gamma^\nu$$
A: I just want to add something to J. Murray's answer that might otherwise confuse you if you have not had an introduction to group theory yet.
The concept of Lorentz transformation is more general than the $4\times 4$ matrix you already know. You can construct an abstract set of transformations that have the same composition properties of those $4\times4$ matrices and this is called the Lorentz group.
Studying the so called representations of the Lorentz group you will see that there are all sorts of different objects with  different transformation properties under the Lorentz group (that  we call scalars,vectors, tensors, spinors, ...).
A  more rigorous statement would be:

A vector is a $(1/2,1/2)$ representation of the Lorentz group.

So until you know group theory just trust your book or your professor and know that there is a precise mathematical meaning behind the  apparently tautological statement "a vector is something that transforms as a vector".
