First, welcome to the SPF, the Friendliest Forum Around [FFA (tm)]. :wave:
Realms play refers to Blizzard's free online gaming service, battle.net - specifically the "closed" portion of the service, where Blizzard keeps all character information on their servers. It is referred to as "the Realms" because Blizz divided it up geographically: US East Realm, US West Realm, Europe Realm, and Asia Realm. Most posters in the SPF do not play on the Realms, although many of us used to.
Open is also battle.net, but is the section where the character information is stored on each client computer (i.e.: your character is on your computer, mine is on my computer, etc.) I think the games themselves still are served from Blizzard's computers, but I'm not 100% certain of that.
You don't mention it, but there is one other Multi-Player (MP) game option: TCP/IP. That is like Open, minus the Blizzard match-making service. An individual hosts the game on their computer, and different people can connect by IP address to the game - which the host must tell them somehow. Each person's character is on their own computer.
Magic Find (MF) has no effect on gambling, IIRC. Your level does affect what affixes it is possible to get, though.