Royal assent is the final stage during the passage of legislation through Parliament.
A bill which has been given royal assent becomes an act of Parliament and is then law.
It is a purely formal stage in which the royal seal of approval is given to the legislation.
In reality the monarch is very unlikely to withhold consent. Queen Anne was the last monarch to do so in 1707.
The monarch does not give assent in person - the last time this was done was in 1854 - instead it is given by the Speaker in the Commons and the Lord Speaker in the Lords.
By ancient tradition, royal assent is given in Norman French in the words "La Reyne le veult", which roughly translates as "the Queen wills it".