One way of doing this is with cell fracture and adding rigid body physics. Add a cube to your scene and Tab into edit mode and press W>subdivide and subdivide it a few times to give it some more geometry. Note that with objects having more complex geometry, you can skip this step.
Next, run the cell fracture add-on and go to your second layer to begin working on the fractured cube.
First we are going to press Bto box select the fractured cube, and select add active in the physics tab of the tool panel on the left. Press T if it's not open.
Next, add a plane below the cube and again go to the tool panel>physics since you need it to react to the fractured cube, click add passive.
Now if you play your animation your cube will just crumble and fall apart. To fix this, go to the physics>rigid bodies tab in the properties editor. You want to adjust the sensitivity margin next.
Play with this until you like how the animation looks.
If for some reason, the cube crumbles and doesn't explode then select one of the fractured pieces, then box select the entire fractured object, and select copy from active in the physics settings in the tool panel and that should fix it. After following these steps you should be able to get something like this or better: