A remake often is also a port, but people ironically seem to call mostly direct ports remakes.
So here's a handy dandy guide, as the guy I'm quoting pointed out there's some confusion up in this bitch:
A port's a direct code translation from one architecture to another. Shenmue 2 on the XBox was a port of the Dreamcast game.
Sometimes you get added content with a port, or a new script, in which case you can called it an 'enhanced port'. This would be the DS version of Chrono Trigger. Or the DS version of Dragon Quest IV, although that was more of a gutted-ass port of the PSX remake. With a bad translation.
If you get a remake, it's implied that the entire game has been redone, in some sense. Like this DQVII remake.
Sometimes, you just get an emulator bundled with a rom of the game which has been hacked to play FMV somehow, such as the PSX version of FFVI.
And on really rare, truly special occasions, you get a box full of musical tarantulas.