A cross-functional team is simply a team made up of individuals from different functions or departments within an organization.
Teams like this are useful when you need to bring people with different expertise together to solve a problem, or when you want to explore a potential solution. For example, you might put together a team made up of people from finance, engineering, production, and procurement to come up with a solution to reduce the lead-time for a new product.
One approach is for team members to be "loaned" full-time to the cross-functional team, returning to their day-to-day role once their contribution has finished. Alternatively, they may work on a part-time basis, continuing with their existing responsibilities alongside their cross-functional team-work.
The most important distinction between the creation of a cross-functional team and the formation of a new department is that members of a cross-functional team maintain substantial links to their day-to-day responsibilities and to managers in their "home" department.