If you've played World of Warcraft (or any RPG) you already know some of the basics. It's a roleplaying game in which you make your own character by choosing their race, class, background, starting spells and equipment. Much like WoW, these choices all have their own pros and cons (Certain races have bonuses to certain stats that might make them better suited to one class over another. Different races also have different racial abilities or perks, as well as different languages they may know how to speak). Unlike WoW, base stats aren't usually preset, and are instead allocated either through dice rolls or a point-buy system. The biggest difference between D&D and a game like WoW is that the story (and the challenge of the gameplay) is entirely up to the Dungeon Master. The game has set rules and guidelines for things, but beyond that the players are at the mercy of the DM, and must use their own wits to solve the problems and encounters thrown at them. This is where D&D gets really interesting. Problems can be solved in roundabout, unexpected ways. Things can happen directly because of a player's actions (or lack thereof).

For example, in one session in particular, our party was exploring a temple that had a bunch of goblins in it (just your run of the mill dungeon crawl). Because of the way our party had made their characters, every single member of the group (barring our Druid, Kaze) had Darkvision, so the temple's dark halls weren't really an obstacle for us. At one point, we had come across a room filled with goblins that was lit by a fire in the center. Now, Kaze wasn't the brightest of druids. Before we entered the room (and alerted the goblins to our presence), Kaze had the "greatest" idea of all time. He uses a spell to put on the fire in the room before immediately turning to us and saying "Now they won't be able to see us coming!". For a moment or two, there was silence amongst the group, before my Sorcerer turns to the rest of the group and asks: "Wait...don't Goblins have Darkvision?" Our DM then asks us all to make an Intelligence check to see if we knew whether or not the goblins had Darkvision, and sure enough everybody failed. That is, everybody except Kaze. The Dungeon Master then simply tells us "Goblins have Darkvision". So our Druid had ultimately put out the only light source in the room (rendering him effectively blind) before immediately remembering that putting out the fire didn't actually affect the goblins. So we proceeded to fight the goblins while our Druid Kaze stumbled around in the dark like an idiot.

As for how physical of a game it is or how rigid or impromptu the story is, it's entirely dependent on the Dungeon Master and the group of players playing. I know there are some Dungeon Masters who like to go as far as to make actual "potions" (colorful drinks like kool-aid or some other kind of juice) for their players to drink when their character drinks a potion in game. Other Dungeon Masters don't use anything but pen and paper. And some groups (like myself and my friends) use computers and online resources such as Roll20 to play (as we're all over the country and can't play in person). Regardless, if you're a nerd who's into MtG or WoW, odds are you'd probably enjoy D&D as well. If you have a group of buddies and the time, I'd highly recommend it, as it's usually a blast (and the stories are almost always hilarious).

TL;DR: Yes it's kinda like "make believe", but it's still very much a game. Sometimes there's a board, sometimes there isn't. Storylines are not really set, depends on the DM. There are rules, but at the same time, those rules are generally guidelines.