Is a normal force acting on an object that is hanging from the ceiling? I've only taken two semesters of calculus-based physics as part of my general education credits at university. Please keep this in mind when reading this question and framing answers.

Today I observed a gecko on a ceiling and began wondering how it managed to stay put. A bit of googling revealed that the magic is actually van der Waals forces, not an unusually high coefficient of static friction or something.
So I tried to draw a mental free body diagram of a gecko at rest on the ceiling. I drew a downwards arrow representing the gravitational force, and an upwards arrow for the van der Waals force. And then I wondered -- is that it? Is the van der Waals force upwards equal to the downwards force of gravity?
When I studied physics at university, we always drew a normal force perpendicular to the surface the object in question was resting on. We were told this represented the force exerted "upwards" by the surface to counteract gravity. By doing this we could calculate the magnitude of an unknown force by having the normal counteract gravity.
In my gecko situation, my unknown force is acting counter to gravity. Is there a normal force at play here too? Or is the van der Waals force taking its place?
 A: Firstly, the 'normal' force is a contact force which opposes two surfaces being forced together.  It is not caused by gravity, but it can be the response to gravity (eg when gravity pushes a book into contact with a table).  It is caused by inter-molecular repulsive forces.
Secondly, the van der Waals force is an attractive force.  This is the only upward force here, pulling the gecko towards the ceiling.
However, does this mean that the pads on the gecko's feet are so finely controlled that the van der Waals force exactly balances the weight of the gecko?  ie If the gecko has a larger breakfast than usual the pads somehow exert a stronger van der Waals force to compensate?  I am not sure this is the case.
Even if the gecko can control the van der Waals force, I doubt that it can be done so precisely.  I think it more likely that the van der Waals attractive force is always slightly greater than gravity, pulling the gecko into the ceiling.  The usual repulsive inter-molecular forces then come into play to oppose the net upward force on the gecko. The normal contact force is reactive and adjusts to provide a perfect balance of forces.
So in addition to its weight (downward) and the van der Waals force of attraction from the ceiling (upward) I think there is also a small additional 'normal' (in both meanings of the word!) contact force from the ceiling acting downward on the gecko.  This is where I disagree with Harm Moolenaar. (I do not understand what Joce is saying - possibly the same as Harm?)
To some extent the issue might be one of semantics - ie whether 'van der Waals force' includes the inter-molecular repulsion or is separate from it.  Nature does not distinguish one force from another, only we do that, for our own convenience.  So I think it is an issue of how forces are distinguished and classified.
A: "Normal" is a mathematical synonym for "perpendicular".   
So the VdW force is a normal force in this case.  The normal on a book due to the table on which it sits is due to the elastic compression of chemical bonds at the surface of the table.  "Normal" describes the direction of the force, not the nature of the force. Similarly, "centripetal" describes the direction of a force whose causative agent might be gravity, or tension in a string, or ...
Note that if the gecko were adhering to a wall that is not horizontal, the terminology gets a little ... sticky.
A: Let's rephrase things slightly.
When an object is resting on top of a plane, it exerts a downward force on that plane equal to its weight (so it is along the inward-pointing normal to the plane). The "unknown force by having the normal counteract gravity" is the reaction force of this plane (along its outward-pointing normal). This is nothing but action and reaction principle (Newton's 3rd law).
This law remains of course true in case an object is adhering below a plane, only the direction of forces are inverted there: through its adhesive contact (Van der Waals forces), the object must exert on the plane a downward force equal to its weight (if it doesn't--that is, if adhesion force and thus plane's reaction force are lower than weight--, the gecko falls down), so along the outward normal of the plane. Therefore, the reaction force of the plane which balances the weight (and allows the object to remain on the ceiling) is along the inward-pointing normal, and its vertical component must balance the weight of the object. This is the adhesion force you have drawn in your representation, which is actually the reaction force you were looking for.
A: Van der waals force takes it's place, no normal forces apply here.
imagine two plates (on which VDW forced do not apply), One in mid air and one pushed to a ceiling. Both supported by a block or something. If you take away the blocks both plates fall equally quickly. If normal force would apply the one pressed to the ceiling would fall more slowly.
