The servers are split up into either a PvP or PvE, right?
Does that mean what it looks like? A PvP server is PvP everywhere, all the time? And PvE is PvP only in the Battlegrounds?
I fairly certain that's the case but wanted to make sure.
Servers are split up into PvP and "regular" (i guess is the best way to put it).
Both types are Horde vs. Alliance servers.
On "regular" servers, you can PvP to some extent if you chose (and a lot of players seem to do this from what I can tell).
I am not an expert on the matter but it appears to work like this. When you attack a faction NPC (mostly NPC town guards) of the other side you get flagged PvP for a period of time. When flagged, the opposite faction can attack you (and in doing so become flagged themselves if they are not already). On a side note, if you cast a benificial spells on your own faction's NPCs, it will turn on your PvP flag.
There are some zones set up that have small (relatively weak in terms of NPC guards) towns of both faction in them. So what happens is a group or several groups of one side will go attack the town of the other side. This will draw out the other side's players that want to PvP and then you get a back-and-forth battle going between the towns.
Also, there are some quests that actually flag you as PvP (they basically force you to slay faction NPCs which turns on your flag). So this also encourages PvP action because you have folks on your side trying to complete the quest(s) while the opposing side has PvPers in the area that know about the quest and seek slay them. I have only seen a few of these types of quests though so I would consider them rather rare at this point.
The PvP servers work similarly except that many of the zones are designated as "contested". What this means is that you are flagged for PvP when you enter the zone. So you can be killed then and there. For "owned" zones, meaning zones that are designated as controlled by your faction, your PvP flag is off but any enemy in the zone is its flagged on. So you have the option to attack them (and then can be attacked) or ignoring them (and thus making it so they can not harm you).
Edit: I think Blizzard is making a mistake in using old naming conventions for the different server types as it seems to be causing a lot of confusion. i.e. lots of new players do not realize that "regular" servers have PvP also just not the the extent of the "PvP" servers.