Two quick things to make clear first:
1. Two blocks are considered 'connected' if they share a face
2. When I refer to 'resource' blocks, I'm referring to ores for pickaxes, logs + leaves for axes, and clay + snow for shovels.

This enchantment has a different effect when used against resource blocks vs non-resource blocks...
For resource blocks:
When it comes to pickaxes and shovels, the chaining radius will be the enchantment level + 1. So a pickaxe with Chaining 3 will cause all connected blocks in a 4 block radius to be destroyed as well.
Due to the size of trees, axes with chaining will have a radius of the enchantment level * 10. So an axe with Chaining 3 will be able to take down an entire jungle tree!

For non-resource blocks:
The effect depends on what the 'Chain Block Mode' setting is set to.
'Resource Only' will only allow chaining to occur on resource blocks.
'Partial Strength' will allow chaining to occur in a radius of the enchantment level - 1. So having a tool with Chaining I will mine non-resource blocks the same as if you had no enchantment (since it will chain on blocks in a radius of 0 blocks...or in other words, just the block you mined). It will still chain on resource blocks though, in the way described above.
'Full Strength' will allow chaining to occur in a radius of the enchantment level.


The config setting 'Chain Growth Mode' determines the shape that the chaining grows in. For a visual, see here.
And for those interested, the shape is caused based on using one of the three ways to determine distances between two points on a grid:
1. Euclidean (Circularly)
2. Manhattan (Diamondly)
3. Chebyshev (Squarely)
Since it'd be a bit wordy to explain how they're calculated, see here for a visual :)