Union and League are two differant codes. They are more of less the same but with slightly differant rules and methods of play.
The Celtic League is a rugby union league in which the provincial teams of Ireland, Wales and Scotland compete. In England they have the Zurich Premiership league.
The British and Irish Lions is a team that is made up of the best players from England, Ireland, Wales & Scotland which does a tour every four years in which they play three test matches against a country. Last time it was in Australia. This year its in New Zealand.
So basically there are two teams, the test team(the better team) which play New Zealand in three tests and there is the second team which plays New Zealand clubs over the course of the tour.