What does the legislative branch do?

The Legislative branch makes laws for the nation. The legislative branch also can override the presidential veto. When the president does not want to approve a law, he or she can veto it, but the legislative branch (Congress) can overrule it by a 2/3 vote of both houses.
In most nations of the world, its legislative branch puts forth bills it votes on to be laws. The proposed legislation is just that, a proposal. Such proposals are normally signed into law by a nation's chief executive. In many nations, not all, the chief executive can veto proposed legislation. If the legislature has enough votes, it can override most vetoes. It should be noted that one party government systems normally agree on a law or policy as the top executives and the law making body are all in the same party.