Unless I'm just messing around on youtube the way I normally listen to music is to sit down with an album and listen to it all the way through. Mostly because it's kind of a hassle to sit there and pick which song you want to listen to next, I guess you can do shuffle, but a good album, even if it's not a concept album, is going to have a cohesive feeling that you just aren't going to get like that. I'm not sure that it's that whole albums aren't given attention anymore, because I can name a few good modern albums (anything by Andrew Bird, Elliott Smith, or Aesop Rock is gonna be really solid, if not excellent), or if it's just that I don't listen to modern music much. But musicians like The Beatles, Hendrix, Bob Dylan, Bert Jansch, and especially King Crimson all put out consistently solid albums, with few, if any, weak tracks.
I think my ideal album is varied, yet still working towards and over-arching theme. The best way to describe it is hearing the musician approach a musical idea from several different styles. King Crimon's first album (what I would rank as possibly the best concept album of all time) is a great example of this. It goes through a few folk-inspired styles, from Ian MacDonald basically just on his flute to the proto-metal of 21st Century Schizoid Man, but not only do the all retain the English folk influences, but they all, not just lyrically, build to a culmination of despair and resignation at society. It makes listening to the same band for a whole album interesting without being repetitive while still making you feel like the album accomplished something. I think, probably as a consequence of the way I listen to music, that repetitive song-writing is the worst thing a band can do, nothing will make me stop listening faster.
There are exceptions though. Dylan's Blood on the Tracks is played in two keys, Open D and Open E, and yet the album's songwriting greatly surpasses those limitations so that every song feels distinct while not really actually being so. But not everyone can write like that, and in anyone else's hands it would have turned out one of the most boring albums ever.