Center of Mass:

Pretty straightforward. To simplify calculations most forces (excluding drag) can be applied to this point, instead of per-part like real world, which is then distributed over to other structures.

The center of mass is dynamic, so therefore it is recommended to put the fuel tanks near the center of mass (and on large aircrafts, usually inside wings). Ground vehicles can almost completely ignore that.
A good way to see how fuel impact the CoM is to build the craft, drain it of fuel, and put in fuel where it had the least impact on CoM. Using fuel as weight to compensate for a bad CoM location is unsustainable (as the fuel eventually get used up). Use dead weight for the cause, or put heavy modules.

Center of Thrust:

In general, not going to be significant except for VTOL craft. Extremely highly recommended to be in-line to center of Mass on the axis of movement (e.g., for a plane, the z of CoT must be the same with the z of CoM, and the y of CoT must be somewhat close to the y of CoM. The location of CoM about x axis is mostly insignificant).
Since most aircrafts are going to be symmetrical, this is usually not a problem. However, the CoT in relation to CoM on the y-axis had some interesting effects:
If the CoT is above the CoM (e.g., wing-mounted engine on a top-wing aircraft), the aircraft will tend to pitch down (because the force induce a torque about CoM). This will largely not be a issue during flight, but it might increase the difficulty of taking off/landing.
If the CoT is below the CoM (e.g., wing-mounted engine on a low-wing aircraft), the aircraft will tend to pitch up (because the force induce a torque about CoM). This will have interesting effects as it can potentially be used to fight against nose-down on takeoff and/or landing.

Simpleplanes stock designer also don't show the direction (which is oftentimes very significant. Imagine a craft flying backwards).

Here comes the fun part.

Center of Lift

But first, what's the "center of lift" without "lift"?
"Lift" in this game is commonly regarded as "upward force generated by flat surfaces", with the aircraft main wing generating the largest share. This is mostly true.
One notion of "lift", however, is that the surface doesn't necessarily have to generate lift when airflow is perfectly parallel to its direction. That characteristic is derived from the difference in curvature on the top/bottom of the craft. It is rather the ability for said surface to provide a force perpendicular to its surfaces.
This is also why some wingless aircrafts are able to fly -- because the airflow striking the lower surface is forced to go down, which generated an upward force. They also flow slower, so the air in the upper section flows faster (in relation), which equal to less pressure, which equals to more upward force.
The reason it doesn't fly well, is because the wings don't necessarily work this way. In real life (e.g., birds), their entire wing twists (they have feathers, remember?) ever so slightly to generate different amount of lift on the left/right, which allow them to roll. Their tail is clearly responsible for pitch. Yaw is not necessary.
This characteristic is consistent with the wright flyers. It's also because of this characteristic that the wright flyer brothers are unwilling to make them bigger -- because they can't.
By contrast, the modern aileron is a completely different picture. It's invented by Glenn Curtiss (and his friends), who are the opponent of wright flyer bros. The aileron is also much more mechanical in nature, simply by increasing the deflecting angle and as thus don't handle as well as the more "natural" approach of the Wright brothers. But their mechanical simplicity allows the motorcycle manufacturers to quickly build larger (and therefore heavier), more powerful, and faster crafts that eventually ousted the Wright brothers.

Since we now know the method of generating lift and the method of controlling lift, we can talk about center of lift.
Why is center of lift so extremely significantly important?
The CoL, as the name suggest, is the center of all lifting surfaces combined. This includes the main wing, but this also include the horizontal stabilizers and/or canards (lifting surfaces mounted before the main wing, respect to axis of movement).

Note: The direction in the below section is in respect to the axis of motion.
Note: The direction of CoL is almost always pointing up. However, in certain cases (e.g., spoiler of a race car), it is pointing down.

If the CoL is behind the CoM, guess what ... the craft will pitch down.
If the CoL is in front of the CoM, guess what ... the craft will pitch up.
If the CoL is to the left of CoM, guess what ... the craft will roll right.
If the CoL is to the right of CoM, guess what ... the craft will roll left.

You might have already started to have cool ideas of how a properly designed aircraft will work. Read on.
Because we now know the effect of the location of CoL respected to CoM is so important, why do we allow it to be off-centered?
Because the center of lift is dynamic. It changes throughout time.
Note: The following section assume a no-wind condition, like the majority of the major portion of most time most people set their Simpleplanes environment variable at.
In a hangar or when on ground, the CoL is negligible (and is the same as you can see in the designer).
When the aircraft pick up speed on the runway, the wings with asymmetrical airfoil start to generate more lift, because the difference in airflow speed start to become significant. This causes the CoL to move toward them.
This is why some craft will have an "auto lift-off" speed above their recommended takeoff speed. Those craft WILL suffer from poor top speed, unless they also have very large horizontal stablizer.
This is also why the IAS (indicated air speed, proportional to the amount of lift gained) of the aircraft will impact how it handles (pitch-wise).
Which means that in theory, the CoL can be anywhere CLOSE to the CoM without having significant impact. It can even be in front of it (remember dynamic changes!).
And it has been done. There are crafts with weird CoL locations, because those crafts have "good computers" that can adjust the deflection rate of the control surfaces dynamically (which, really, simply improves stability). Pushing (moving forward) on the stick and fly manually can be done, but it will probably be very exhausting as the pilot will need to be highly focused.
Manipulation of the location of CoL can be done by changing the amount of lift generated by a wing. This can be done in a few ways:
1. Change the angle of the wing in relation to the airflow
2. Change the geometry of the wing
3. Change the size of the wing

The angle of the wing can be changed via rotators. Certain fighter aircraft is seen to have them (perhaps for increased maneuverability).

The geometry of the wing can be changed or via ailerons (or flaps).

The size of the wing can be changed during flight by extending lifting surface outward
How do you do that?
Well, if you had been to an airport before, chances are that you had seen the big jets with their massive flaps. Those flaps extend front (and back), and they curve down a bit, effectively making them bigger wings. Those flaps are called "fowler flaps". (so technically they are flaps)

The gaps are there to reduce wake (turbulence), effectively making them mini-sized wings attached behind and below the main wing.
So now you know most of the physics-related mess behind making airplanes (heavier-than-air crafts with sustained atmospheric flight capabilities, as scientific as it goes), I expect you to exploit them to make your existing crafts better.

Note:
As much as it might be pleasing to set up the wing geometry so the plane auto levels (pitch 0), it might be undesirable as this may lead to unwanted stalls due to insufficient lift. Some planes are incredibly difficult to stall (except those big planes and on windy days), and will only do so when their controls are pushed to the extreme (and even if it is done, the craft might simply nose down to prevent that from happening). A roll-leveling craft (somewhat difficult) is always welcomed.
Note:
The amount of lift generated by the wing is proportional to the angle of attack. The angle of attack is the angle between airflow and the wing. Lift increase linearly as the angle increase, but at a critical angle, turbulence occur on top of the wings instead, decreasing the amount of lift. The point when the lift first starts decreasing (due to a excessive AoA) is called critical angle. It depends on wing geometry.

Note:
Wing sweep changes the center of lift as well as decreasing drag (by making the air move non-perpendicular to the wing's edges), which "makes the wings think they are flying slower". This mechanism might have not been simulated in-game.