In this paper we characterize the compactness of the commutator
$\left[ b,\,T \right]$
for the singular integral operator on the Morrey spaces
${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$
. More precisely, we prove that if
$b\,\in \,\text{VMO}\left( {{\mathbb{R}}^{n}} \right)$
, the
$\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$
-closure of
$C_{c}^{\infty }\left( {{\mathbb{R}}^{n}} \right)$
, then
$\left[ b,\,T \right]$
is a compact operator on the Morrey spaces
${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$
for
$1\,<\,p\,<\,\infty $
and
$0\,<\,\lambda \,<\,n$
. Conversely, if
$b\,\in \,\text{BMO}\left( {{\mathbb{R}}^{n}} \right)$
and
$\left[ b,\,T \right]$
is a compact operator on the
${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$
for some
$p\,\left( 1\,<\,p\,<\,\infty \right)$
, then
$b\,\in \,\text{VMO}\left( {{\mathbb{R}}^{n}} \right)$
. Moreover, the boundedness of a rough singular integral operator
$T$
and its commutator
$\left[ b,\,T \right]$
on
${{L}^{p,\lambda }}\left( {{\mathbb{R}}^{n}} \right)$
are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right.