Dual Member or mixed-member electoral systems are a hybrid of two electoral systems, first past the post or some other constituency system, and the list system. A Parliament or Assembly elected this way has two classes of members, though both usually have the same voting and other rights.
Some members are elected by individual constituencies, and other members are elected by a list system. But the list is used to top up parties that received fewer than their fair share of seats via the constituency process. So if half the members are elected by the list and half by constituencies, then a party which received a tenth of the vote but didn't win any constituencies would get a fifth of the seats being decided by the list election.
Traditionally this has been the West German electoral system, subsequently used by the whole of Germany. Germany also combines it with a threshold that any party must get 5% of the national vote to get any seats on the list, that's to keep our Nazis and other extremists. In recent years it has also been used for elections to the Welsh Assembly and the Greater London authority.
It has the advantages of achieving both the fairness of the list system and the constituency link of the First past the post system. Its disadvantages include, larger constituencies, and the possibility that a party can do so well in the constituencies that it loses some members it was expecting to have elected from the list. British Liberal Democrats and other supporters of preferential systems don't particularly like it because it doesn't give voters a choice between candidates of the same party, but they along with most progressives think that unlike first past the post it is at least sufficiently proportional to count as democratic.
It can either be done by giving electors one ballot paper where they vote for a candidate from a party and their one vote counts for both parts of the election, or it can be done as in London by giving each voter two ballot papers, one for their constituency and the other for the top up list. The first system has the disadvantage of putting those who want to vote tactically in a dilemma, the second has the disadvantage that an elector can game the system by voting in the constituency for a party that is likely to win in that election and then vote in the top up for a smaller party which will get all its seats that way.