"A shared competence" would normally entail that two different decision-makers share the right to decide and make laws.
When the EU and its member states share a competence, the member state loses its 'competence' (power to take decisions) when the EU decides to regulate.
So EU legislation does not only replace the content of a national law, it also removes the national right to legislate in the covered area.
The Lisbon Treaty Art. 4.1 TFEU defines shared competences to be openended between the exclusive competences and the supporting actions. New areas can be added:
1. The Union shall share competence with the Member
States where the Treaties confer on it a competence
which does not relate to the areas referred to
in Articles 3 and 6.
2. Shared competence between the Union and the Member
States applies in the following principal areas:
(a) internal market;
(b) social policy, for the aspects defined in this Treaty;
(c) economic, social and territorial cohesion;
(d) agriculture and fisheries, excluding the conservation
of marine biological resources;
(f) consumer protection;
(h) trans-European networks;
(j) area of freedom, security and justice;
(k) common safety concerns in public health matters,
for the aspects defined in this Treaty.
3. In the areas of research, technological development
and space, the Union shall have competence to carry
out activities, in particular to define and implement
programmes; however, the exercise of that competence
shall not result in Member States being prevented
from exercising theirs.
4. In the areas of development cooperation and humanitarian
aid, the Union shall have competence to
carry out activities and conduct a common policy;
however, the exercise of that competence shall not
result in Member States being prevented from exercising
See also Categories of competence and Convention working group on Complementary Competence.