Apple's Videos app was first introduced with the original iPod touch. The original iPhone shipped with an iPod app that bundled music and video together but an iPod app on an iPod device must have made less sense. Later the original iPad launched with a Videos app as well and, eventually, the iPhone fell in line. Now all iOS devices ship with a Videos app that lets you watch the movies, TV shows, and music videos you've bought from the iTunes Store.
Better yet, thanks to iCloud and re-downloading, you no longer have to worry about syncing content over from the desktop, just tap the cloud icon and pull it straight down to your iPhone or iPad.