What does it mean if something Transforms Covariantly ? (Quantum Field Theory)

What does it mean if something "Transforms Covariantly"? (Quantum Field Theory)

Referring to an isospinor, [itex]\psi[/itex] which transforms as [itex]\psi[/itex](x)→ [itex]\psi[/itex]'(x)=S(x) [itex]\psi[/itex](x) (S(x) being an n by n matrix)

I'm told that it is clear that [itex]∂_{μ}[/itex][itex]\psi[/itex] does not transform covariantly.

Now, correct me if I'm wrong, but it would appear that [itex]∂_{μ}[/itex][itex]\psi[/itex] ' can be found by the product rule to be S([itex]∂_{μ}[/itex][itex]\psi[/itex]) + ([itex]∂_{μ}[/itex]S)[itex]\psi[/itex].

What is meant by that it doesn't transform covariantly?

I know what covariant vs. contravariant indices are, but I don't know what it means for something to transform covariantly. I understand that we *want* a covariant derivative, but I don't understand why =|

Any and all help would be great!

**from Ryder, in case anyone was wondering