Falcon 9 (not heavy) has a first stage and a second stage. The first stage consists of one kerosene tank, one oxygen tank and nine engines. This is also called a core.

Falcon 9 heavy (no crossfeed) has a first stage and a second stage. But the first stage consists of 3 cores that run in parallel. All three cores start and stop at the same time. So 27 engines start at the pad.

Falcon 9 heavy with crossfeed has sort of two or three stages, depending how you count. Outwards it looks very similar to the no-crossfeed Falcon 9 heavy. The topmost stage is similar to the two already mentioned configurations. The three cores look pretty similar. All three cores start at the pad (27 engines running), but the propellant for the center core's engines is taken from the outer cores' tanks. This means the outer cores run out of propellants quite quickly. They can then be discarded in flight, and the center core starts using propellants from its own tanks. Now it can fly without the dead mass of the outer cores and as a result the rocket can carry a bigger payload.

Why start the core engine at the pad, and not make it a real second stage that starts up only higher? Because then you get more thrust, the engines are not dead weight during early flight.

Some rockets like Titan IV had a somewhat similar parallel configuration but they did not start their center engine except at high altitude.

Right. OK. So with a simple 2-stage rocket like the Falcon 9 you have the first stage which has the 2nd stage stacked on top of it which has the payload stacked on top of it. During a normal flight the first stage (which is the biggest stage) lights, lifts the whole thing off the ground, and pushes the rocket up into the air and starts getting it up to speed. Then the first stage runs out of fuel, it separates and then the 2nd stage takes over, pushing the payload the rest of the way to orbit.

The Falcon Heavy uses the same "2nd" stage but it uses 3 first stages instead of just one, which are called "cores". Each core is just a Falcon 9 first stage, for the most part. The idea of the Falcon Heavy (similar to the way the Delta IV Heavy works) is that you have three times as many first stages pushing, which leads to increased payload.

With the cross-feed option you end up with effectively a 3-stage launcher. The outer two cores becomes like the first stage, and they are jettisoned when empty, then you have the inner core which now becomes a 2nd stage, and the previous 2nd stage is now a 3rd stage (or just "upper stage", to avoid confusion).

But, as mentioned, when you fly this way the 2nd stage core ends up way faster and way farther away when it's done thrusting, so in the prospect of having it keep around some extra propellant so it can thrust back to the launch site and land so it can be reused becomes much more difficult. But in a worst-case scenario all of the "cores" can be run at the same time, so they burn out the same distance away from land and can each return and be reused (saving a lot of money). You end up with only a 2 stage vehicle (3 1st-stages all running at the same time plus the upper stage) which would decrease performance, but because you recover all of the first stage cores (instead of only 2/3 of them) the cost benefit would probably be worth the payload reduction.

Wow thanks, between this and Gravityloss's comment, I'm really starting to get a clear picture.

So on the Falcon heavy, is the plan for all three cores to return to the launch pad independently? Just landing a few hundred feet away from each other?

Pretty much.

Also, I missed a reusability scenario with Falcon Heavy. You use crossfeed, you return the outer 2 first stage cores, and you dump everything else. This is more expensive than reusing all 3 1st stage cores, of course, but it provides a considerably amount of payload for the cost of essentially an expendable F9 plus the use of the reusable boosters. So you'd get very heavy lift capacity (I dunno, maybe 30+ tonnes to LEO) at well under $100 million, which in the current launch market is a crazy price.