I've mulled this over a number of times. Here is what helps me play better with a knight. I offer it not as a general solution, but as a solution that has worked for me. I hope you find it useful:
- Understand how the Knight moves COMPLETELY.
- Understand ALL the Knight threatens COMPLETELY.
- Understand that the value of the Knight relative to the Bishop may change.
- Understand by colour, what the Knight threatens when he sits still or moves twice (Rule of Even).
- Understand by colour, what the Knight threatens once he moves (Rule of Odd).
Point 1. Most people know how the Knight moves described as a "L". I've found this to be inefficient since this is only visualizing one of his possible moves and it's possible to miss some of his possible moves thinking this way.
To truly see his potential a player needs to know all of his possible moves at once. Here's how I visualize it (and will try to explain clearly how I see it):
Around the Knight is a box the corners of which match the colour he is sitting on. The Knight jumps outside of the box to the opposite colour squares touching the corners.
It is possible to look at a board and see this "box" while playing and see all of Knights potential moves at once, rather than just one. The ability to see this box rather than a single move is something of an epiphany.
Point 2. Understanding all the Knight threatens at once, rather than a single threat allows the player to understand all the pieces he can attack.
Just as above (Point 1) where seeing all of a Knights moves show his true potential, seeing all he threatens shows all of his attacking potential. Many players identify what he threatens (in a somewhat sequential manner) one threat at a time. However, this is inefficient and may inevitably miss some. The ability to visualize his box allows a player to see all he threatens which allows a player to realize his full attacking potential.
Point 3. Some theories of Chess don't assign the Knight a static value relative to the Bishop (meaning the same unchanging value). Consider this for a moment:
If the Bishop works best on an open board with far reaching open diagonals BUT the Knight can jump around pieces on a crowded board but whose reach is always limited, it stands to reason the Bishop's value increases over time as the board opens up and as the game goes on; whereas the Knight's value decreases overtime since the board opening up removes his advantage. Accordingly, consider assigning the Knight a value of 3.5 in the beginning of the game (Bishop at 3.0), and flip these values as the game goes on as it's appropriate.
Point 4. An un-moved Knight threatens the opposite colour he's sitting on (or if he moves twice).
A Knight on BLACK is threatening WHITE. A Knight on WHITE is threatening BLACK. Remember this and consider it when considering position.
Point 5. A moved Knight threatens the same colour he sat upon before moving.
A Knight moving off BLACK will threaten BLACK once he comes to rest. A Knight moving off WHITE will threaten WHITE once he comes to rest. Remember this and consider it when considering position.
These last 2 points really help in end-game situations where you're forced to chase an odious opponent around the board. If you want to threaten the same colour you sit upon, you must move. If you want to threaten the opposite colour, stay or move twice.