First cousins are two children who share a set of grandparents, but have different parents. Double first cousins share both sets of grandparents, but have different parents.
Let's state that in a different way: double first cousins happen when two siblings of one family marry two siblings of another family. When you and your brother marry Bob and his sister, your children and their children will be double first cousins.
Many states in the USA allow first cousins to marry*; but as far as I can find only one, North Carolina, will allow cousins to marry but will not allow double first cousins to marry. This distinction is an important one. While first cousins usually share 12.5% of their genes, double first cousins share 25%. This is the same, genetically speaking, as being half-siblings, the marriage of which all states prohibit.
If two sets of identical twins marry, the children will be genetically equivalent to siblings, but would still be considered double first cousins, both in common language and legally speaking.
If you would like to learn about double second cousins, double first cousins once removed, and other very complex relations, http://www.genetic-genealogy.co.uk/ has diagrams to explain it all.
* About 23 states allow first cousins to marry, although many require genetic counseling, or that they be over a certain age, or that they can prove that the couple is sterile.