iOS: VLC, one of the best media players for the desktop, has had a long and tumultuous history, but it's back today after being rebuilt from the ground up to be fast and flexible, play virtually any video file, and so Apple won't give it the boot.
The new version is definitely fast, and makes use of multi-core video decoding. It supports every file format that VLC on the desktop does, so you really can play anything through it with or without transcoding it first. The new VLC also supports integration with Dropbox, gives you the option to upload video via Wi-Fi, and even hooks into the iOS Share menu to let you quickly push a video file you've downloaded right into VLC. There's even support for closed captioning and subtitles, and for AirPlay playback so you can push video to the big screen.
Android users, don't fret: You've been stuck with the now year-old VLC beta, but the VideoLAN team says an update on that front is on the way soon. iOS users can hit the link below to grab the new version.
Note: Some people have reported issues with the app store link. If it's not working for you, check back and try again later. According to this thread at Hacker News, the app is available internationally with no restrictions, so it should be in the app store wherever you are.