Mame4droid is a port of the Multiple Arcade Machine Emulator (MAME) for Android devices. The specific version 0.139u1 is critical. Unlike PC emulators that auto-update, MAME for Android is often "frozen" to a specific version to maintain performance on mobile hardware.
Roms, short for Read-Only Memory, are the digital versions of arcade games that can be played on emulators like Mame4droid. In the context of Mame4droid 0.139u1, Roms refer to the game data that is required to play specific titles on the emulator. Roms can be downloaded from various sources online, but be aware that downloading Roms for games you don't own may infringe on copyright laws.