Lorentz invariance? What exactly is meant by Lorentz invariance?
Is it just an experimental observation, or is there a theory that postulates it?
What quantities do we expect to be Lorentz invariant?
Charge? Charge densities? Forces? Lagrangians?
 A: 
What exactly is meant by Lorentz invariance?

A physical quantity that is unchanged by a Lorentz transformation, i.e., a coordinate system (or frame) independent quantity that is independent of the spacetime coordinates.  Another term used is Lorentz scalar.
Often Lorentz invariant quantities are prefixed with the word proper:  proper time, proper distance, proper acceleration.
Sometimes, they are prefixed with the word invariant, e.g., invariant mass.
This is in contrast with Lorentz covariance.  An equation is said to Lorentz covariant if the equation holds in all inertial reference frames, i.e., if the equation is valid in one inertial frame, the Lorentz transform of the equation is valid.
Other Lorentz covariant objects are four-vectors (and their duals) and higher order four-tensors.

Update to address a comment:

"a quantity that is independent of the spacetime coordinates." that is not right. $\mathbf E \cdot \mathbf B$ is dependent on the
  spacetime coordinates, yet it is Lorentz invariant.

No, $\mathbf E \cdot \mathbf B$ is not coordinate dependent though it can vary from event to event in spacetime.
I'd like to address this comment so there won't be any confusion of the phrase "independent of spacetime coordinates".
To say that a quantity is independent of spacetime coordinates is not to say that it is independent of spacetime.
Consider a scalar field $\phi$ on spacetime.  $\phi$ is a rule that assigns a number $\phi(\mathcal P) $ to each event $\mathcal P $ in spacetime.
Clearly, the number $\phi(\mathcal P)$ is coordinate independent; $\phi(\mathcal P)$ does not depend on the coordinates we choose to assign to the event $\mathcal P$
However, coordinate independence does not mean that $\phi(\mathcal A) = \phi(\mathcal B)$ for two distinct events $\mathcal A$ and $\mathcal B$.
Thus, while the numbers may change from event to event, no choice of coordinates can change the number associated with a particular event.
A: Lorentz invariant is a short-hand for "invariant under action of the Lorentz transformation". It is used to classify quantity that has value that is the same in all inertial frames. For example, electric charge of any electron, or the quantity $\Delta x_\mu \Delta x^\mu$ of any two events.
