Original "fat" Xbox 360 hard drives (grey enclosures) are physically and electronically incompatible with Xbox 360 Slim and E models (black enclosures). You cannot simply swap them between generations.
Purpose-built Slim/E hard drives (250 GB, 320 GB, 500 GB) are the plug-and-play option. Buy new or certified refurbished to avoid counterfeit units.
Any standard 2.5-inch SATA HDD or SSD (up to 2 TB) works when mounted in a compatible third-party caddy. The console will prompt to format the drive on first insert. SSDs dramatically reduce load times and texture pop-in.
Up to 2 TB of USB flash drives or external hard drives can be formatted for Xbox 360 use. This is the easiest route for game installs, saves, and profiles — no disassembly required.
Use the Xbox 360 Transfer Cable (official accessory) or the built-in USB transfer tool in System Settings to move saves, profiles, and installed games from an old drive to new storage.
Important: Compatibility is one-way impossible. A Slim/E drive cannot function in an original Xbox 360 either.