This is a possible answer to the *title* of your question, it is *not* intended to interpret $\dim_{k(x)}(\Omega_{X/Y}\otimes k(x))$ (which is already explained in previous responses).

On a Noetherian scheme, dimension of a coherent sheaf could mean dimension of its support. This is in analogy with the affine case. If $R$ is a Noetherian ring and $M$ is a finitely generated $R$-module then $\dim M=\dim\big( R/\mathrm{ann}(M)\big)=\dim\mathrm{Supp}\ M$, since $\mathrm{Supp}\ M=V\big(\mathrm{ann}(M)\big)$.