A pseudovector is an object that, like a vector, has a magnitude and a direction, and can be written in coordinates relative to a chosen set of coordinate axes, and behaves like a vector when the physical system is rotated; but, upon reflection or inversion of the physical system, the pseudovector behaves differently from a vector.

The most obvious example of a pseudovector is angular velocity. Angular velocity, usually written as a vector, does indeed have a magnitude and a direction. However, under reflection or inversion, it behaves differently from linear velocity, which is a true vector. To see this, consider the following diagram [source]:

The car on the left is driving away from you, so, when you work out the direction in which the wheels are turning, you see that the angular velocity points to the left. Now imagine you reflect the car across the plane indicated by the dot-dashed line. The angular velocity still points to the left.

Now consider a pedestrian jogging, with velocity to the left. Under reflection, the pedestrian is now moving to the right, so the velocity now points right.

Therefore: the linear velocity always undergoes a reflection when a physical system is reflected, but the angular velocity does not. The angular velocity does not behave like the linear velocity (a true vector) under reflection. That's how you can tell that it's actually a pseudovector.

More precisely, under a reflection or inversion, a pseudovector always undergoes an additional inversion compared to a vector. In the example above, to determine the image of the angular velocity under reflection, first you have to reflect it like a normal vector (so it now points to the right) then you have to reverse all three of its components (making it point to the left). This additional inversion distinguishes pseudovectors from vectors.

All pseudovectors in classical mechanics are derived from applying the right-hand rule, in the from of a cross product or a curl. The quantities they represent are naturally described by rank-2 antisymmetric tensors, which masquerade as vectors through Hodge duality---but the Hodge duality taints them, so they end up as pseudovectors rather than vectors. For more mathematical details, see: Brian Bi's answer to How is right-handedness ensured for coordinate systems in dimensions greater than three?

We can quickly enumerate the most common examples of pseudovectors by considering when the right-hand rule is used:

• Angular velocity
• Angular acceleration
• Angular momentum
• Torque
  
• Magnetic field
• Magnetic dipole moment

In contrast, the following quantities are true vectors:

• Linear velocity
• Linear acceleration
• Linear momentum
• Force
  
• Electric field
• Electric dipole moment
• Magnetic vector potential
  

It is a good exercise to convince yourself that this classification is correct for the examples in electrodynamics, by picturing charge and current configurations and then reflecting them or inverting them.