The Casimir force is an effect that arises due to the ground state energy of the quantized electromagnetic field. Conceptually it works like this:
Each frequency mode of the electromagnetic field is treated like a quantum harmonic oscillator. So,therefore, each mode has a non zero ground state energy.
Now, let's place two conducting planes in our space. The electromagnetic field has to satisfy specific boundary conditions in the area between the plates, which alter the number of modes that are allowed to exist in that area. This changes the ground state energy of the field in a way that depends on the spacing between the plates.
Working through the calculation shows that the ground state energy gets lower as the plates are moved closer together. Since the system will want to minimize the total energy, this results in the attractive Casimir force between two conducting plates.
Of course, the calculation is a bit more complicated than this description makes it seem. Technically, the ground state energy of the electromagnetic field is infinite, so a regulator needs to be introduced, but conceptually it works as described above.
Some references for more details if you are interested:
The wiki article on the Casimir force actually runs through an outline of the calculation for two plane conductors.
http://en.wikipedia.org/wiki/Casimir_effect
Also, Tony Zee's "QFT in a Nutshell" Chapter I.9. This is a short overview of the origin of the Casimir force at the level of a senior undergraduate who has had a course in quantum mechancs.
I should note that the Casimir force can also be explained in terms of virtual particles. However, I am not as familiar with that approach, so I'll let someone else give the details.