The most famous of these is clearly "The Wall" by Pink Floyd (who has made a career of concept albums) with The Who's "Tommy" being another strong favorite.
Some, incorrectly, call these magnus opi :) as rock operas. Sorry, this is not right. A rock opera would be more appropriately "Jesus Christ Superstar" or "The Rocky Horror Picture Show" (both of which is not great music nor compelling story lines). In other words, with a rock opera the performers actually take on the personas of the characters where in a concept album the performers tell the story without "being" the characters.
The concept album takes many forms. Technically, the story or idea should be threaded throughout the entire album. So while Rush's "2112" or "Hemispheres" may not technically be concept albums but rather epic songs (aprox. 20 minutes each) but since they would take up the entire side of an LP, I would still call these concept albums.
While the concept album is rarely religious in nature, the idea of creating a long form musical program that tells a compelling story comes from this rich tradition.
For more examples of modern concept albums, check out the following:
Dream Theater's Metropolis Part 2 Scenes From a Memory
Dream Theater's Six Degrees of Inner Turbulance
Neal Morse's Sola Scriptura
The Mars Volta De-loused in a Crematorium
Porcupine Tree's The Incident