Blender Version: December 12, 2007 compile (svn experimental)
Tutorial Version: December 12, 2007
Resources: none
Credit: This tutorial is entirely by LGuillaume. LGuillaume's original tutorial (With minor updates by myself).
1. Explosion of a Sphere.
1_1. Setting up the Scene
To begin, I will show you how to setup a simple explosion of a sphere. To accomplish this, we will use a Plane with 'Deflection' and 'Kill'
enabled and an UVSphere with a particle system.
Delete [x] the default cube, add a Mesh > Plane, and scale the Plane by a factor of 10. (We want the Plane to be big enough to "catch" all the particles).
Note that the new version of Blender adds objects aligned to the Global co-ordinate system, irregardless of which view you are currently in,
and also remains in Object mode when the object is added.
If you would prefer to return to the previous behaviour, activate the buttons 'Switch to Edit Mode' and 'Aligned to View' beneath the
'Add new objects:' header located in the Information Window > [Edit Methods] section.
With the Plane selected, switch to the Object Buttons Window [F7] > Object Subcontext > Deflection panel, and enable Deflection so that
the particle system will detect when a particle touches the Plane.
In the Deflection panel options, you will notice a Kill button - enable it. When Kill is enabled, all particles that touch the Plane will be killed.

Add a Mesh > UVSphere, and move it 4 Blender Units (BU) along the +Z axis so that it is located above the Plane.
Create a new particle system for it in the Object Buttons Window [F7] > Particles Subcontext > Particle System panel.
Since a default UVSphere has 1025 faces, set the Amount of particles to 1025. Also, below the 'Emit From:' text, enable Random and Even, and also set the distribution to Random.
Set the Start frame to 1, to begin the creation of particles at the first frame, End frame to 50, and Life to 100:

If you press Alt+[a] to playback the particle animation, you will see that the particles are created, but they don't move.
To correct that, we will add some Physics to the particles:
In the Physics panel, enter a value for the Normal button (outlined in blue in Figure 1_3 below) to give the particles some speed along the normals
of the Sphere faces (tangent to the face). You can also enter a low value for Random, if you wish to add a bit of randomness to the explosion.
To cause the particles to fall to the ground, adjust the AccZ button (outlined in green) and set a negative value.
To add a bit of rotation to the particles, set the Angular Velocity to Random (outlined in red) and set a value.
TIP: To see the rotation of the particles, you can change the way the particle is displayed in the 3DWindow by setting the Visualization panel option to Axis.

2. OK, but my Sphere does not explode
If you playback the particle animation now, you will just see the particles falling on the Plane.
To add the explosion effect, switch to the Edit Buttons Window [F9], and in the Modifier panel, click the [Add Modifier] button and select Explode.
Enable Split Edges to produce a better effect.

To finish up, we need to return to the Object Buttons Window [F7].
In the Visualization panel, change the option to None and press the Render emitter button, so that the particle emitter will be rendered.

If you playback the particle animation now, you should have an explosion of your mesh.
TIP: You can triangulate the UVSphere (Ctrl + [t] in Edit mode) to give the exploded pieces a more shattered glass appearance, as opposed to the current "confetti" type look.
3. Multi Explosion of a Sphere
Now I want the explosion to occur in 2 stages. In the first stage, the sphere explodes into large pieces and in the second stage, explodes into
smaller pieces.
For the current particle system, change the Amount of particles to 20 and change the End frame to 10:

At the top right of the Particle System panel, there is a button labelled < 1 Part 1 >.
Press the right arrow and then click the Add New button to add a new particle system.
In the second particle system, Start the simulation at frame 15 (the first system stops at 10), set it to End at 30, and make Life: 100.
Make some changes to the remaining parameters, similiar to what we did for the first particle system.
In this instance, we don't use the Normal parameter in the Physics panel, but instead use the Object parameter (outlined in green below) so that the resultant particles will "inherit" their starting velocity from their first generation counter part.

Don't forget to enable another Explode modifier in the modifier panel for the second particle system.

I hope this will help you a little to understand the new particle system.
Until next time, Happy Blendering.