iMore.com has frontpaged a thread (it's HERE) asking about the lack of playing non-YouTube / non-podcast videos in the background; that is, only to listen to the sound but not watch the video itself.
As usual, the exact answer can be found in my articles and full roundups. (No wonder I keep stating my full roundups answer almost all practical questions one can come up with!) For example, this question is answered in the already-public chart (it's HERE) of my forthcoming iOS Multimedia bible. The “Audio playback in background?” row lists the apps that can be run in the background to play back the audio track of videos:
ProPlayer: enable it under Settings / Advanced / Audio Multitasking
HD Player Pro (used to be ATK Player by Khoa Tran Anh): enable it under Settings / Enable Multitasking
OPlayer HD by olimsoft: enable it under Settings / Video Settings / Pause in Background
BUZZ Player HD by BUGUN Software
AVPlayerHD by EPLAYWORKS.Co.Ltd: enable it under Settings / Use Multitasking
and also my personal favorite, GoodPlayer by Hustmobile: enable it under Settings / Background Audio
NOTE: except for ProPlayer, in all the listed players, you MUST disable hardware decoding if you try to play back standard MP4 / MOV files with H.264 video (that is, the most common configuration). Most of the players come with hardware decoding disabled; some, however, don't. The latter is (all this info can also be found in the chart; see the “Hardware playback?” row in the “MP4/M4V/MOV Container support” group), BUZZ Player HD, in which you can disable acceleration by selecting Settings / Playback / BUZZ Player Engine.
This, again, only applies to hardware decoding. Software-decoded videos (almost everything non-MOV/M4V) will always play in the background, regardless of the hardware acceleration setting.
ProPlayer is a different animal hardware acceleration-wise: playback also works with hardware decoding (which, by the way, can't be disabled for H.264 video). The only disadvantage of using ProPlayer is that playback starts from the start of the video every time you (after switching back to it) send the app to the background. (It won't be an issue if you don't switch back to it – that is, foreground it - at all.) That is, if you encounter stuttering audio or other problems using the other players using their software decoder and can live with the restarting bug, give ProPlayer a try.
Playback in a non-fullscreen video pane?
In the linked thread, another common (and unanswered) question was whether it's possible to play a video in one part of the screen (in a pane/window) and enter text in another – as you can also watch YouTube videos while, say, reading comments or browsing the list of the related videos. Sure you can! - but, currently, none of the currently available players. Programmatically it's entirely possible and I'm absoutely sure it's not against the AppStore rules either. I'll come up with an app like this hopefully this night, after having finished writing an article on the effects on the true resolution of upscaling Full HD video to the 2*XGA screen of the iPad 3.