What is a quadrupole moment of a molecule and how does it arise?
This explanation is geared at someone with one year of chemistry. It captures the gist but is not rigorous:
To test for monopoles, dipoles or quadrupoles in a molecule or ion, assign charges or partial charges to each atoms. For partial charges, δ+ and δ- is fine (if you want estimated values for partial charges of simple molecules, use a tool like MolCalc).
To test for monopoles (ions), add up all the charges. If they are non-zero, it's an ion.
If the net charge is zero, test for presence of a dipole: If you can draw a line through the molecule such that more negative partial charges are on one side of the line and more positive partial charges are on the other side of the line, it has a dipole moment. An example for water is below.
If it does not have a dipole, you can check for a quadrupole moment: If you can draw an "X" through the molecule such that more negative partial charges are in opposite quadrants and more positive partial charges are in the other two quadrants, it has a quadrupole moment.
The animated GIF shows the model of carbon dioxide rotated in the presence of such an "X" (or "+" in this orientation if you will). In one orientation (where it is diagonal), the negative partial charges of oxygen are in opposite quadrants, while the positive partial charge is in the center (neither quadrant), showing that it has a quadrupole moment.
For a definition of the quadrupole moment (i.e. how to calculate it from a given charge distribution), take a look at this section of the wikipedia article on multipole expansions, which I found when reading the answer to Can nonpolar molecules exhibit dipole-dipole forces?.
How is it measured for a particular molecule?
The accepted answer for this question cites a paper describing one method.