What is realm time?

It's the time zone that the server is in. Blizzard has datacenters with WoW servers in them all over the US, and they give realm time as the local time in that time zone. All servers for a given realm are in the same datacenter, so time is consistent within each realm, but not across realms.

Simply put realm time is the timezone in which the server is operating in, and is generally based on which battlegroup your server belongs to, and as long as you do not move, your home realm time will stay the same relative to your local time (though it may become an hour or so off around daylight savings time switches).

That being said, you may notice that the realm time changes when entering instances, battlegrounds, or certain parts of the game world. This is because the entirety of the game is not housed on a single server, but rather multiple servers, and when you switch to these other "zones", Blizzard's servers will likely try to pair you with a server that has relatively lower traffic to reduce the hit on game performance (though this is likely the case, Blizzard has kept the specifics on their servers relatively tight-lipped).

What this all means is that while you may be located on the east coast, your home server could be in central time, which would mean a 1 hour difference in time. When you enter a battleground or dungeon, the server with the least amount of traffic may be one on the west coast, so you will notice a three hour difference.

In reality, you should only ever have to worry about your local time relative to your home server's time, as most guilds will schedule things based on server time.