The first song I thought of that has affected the worship culture is a no-brainer: “In Christ Alone” by Keith Getty and Stuart Townend.
Fittingly, for a choir guy like myself, I discovered it first as a choir arrangement. It was only later that I became aware of the larger catalog of new hymns which those two and others from the British Isles were creating for the church at worship. Though it is possible to connect the growing popularity of those new hymns with the general boom in Celtic music (where would PBS pledge drives have been without Riverdance?), I think the more telling reason is how wonderfully those hymns bridge the gap between contemporary and traditional mindsets in the evangelical church.
These new hymns work well with modern instrumentation, traditional instrumentation, folk instruments, and/or orchestral instruments; the melodies are singable and easily learned thanks to their roots in the folk hymn tradition; and the texts are full of fresh language and theological substance. These writers and others like Bob Kauflin helped open the door to many other creative people and groups to offer us new hymns, old hymns with new tunes, old tunes with fresh arrangements, and in general I think have raised the bar as far as the depth of contemporary music for the evangelical church.