When you first start learning chess, one of the first things that you learn are the points for each piece. Having more points than your opponent doesn't guarantee victory (especially when first starting), but it will help you during the game. As you get stronger, it starts to matter so much that even being up a pawn is enough for you to win the game! But then, there comes a changing point where it is not so clear again.
In endgames, it is very often that you have the same pieces as your opponent, but perhaps there is an imbalance with the pawns, even simply the fact that some pawns are just weaknesses! For these more concrete endgames, there are basic principles that you should always keep in mind, but with piece imbalances, the line between the main endings blurs, and it is hard to tell what exactly to follow.
To work around these endings, you must use your own judgement to form a plan. Often, in these endings, it is all about the passed pawns, but sometimes it may also revolve around king safety, and sometimes even both!
But, as usual, there are still going to be basic rules that apply to all of these:
EXPLOIT ENEMY WEAKNESSES
Create passed pawns
BE MORE ACTIVE THAN YOUR OPPONENT
Using the King?
As you can see, there are only really 3 rules I have put down for these endings. It may be a bit vague, but you should definitely not miss out on trying to apply these when playing for a win. You will notice that this time I did not mention USE THE KING, and that's simply because it very much depends on what pieces are on the board. After all, if there are Queens on the board with other pieces, high chances are that you could get checkmated if you dare use it!
Also, in many cases, the pieces themselves are perfectly capable of carrying out the plans on their own, and bringing the king may take a bit too long and leave enough time for them to get enough counterplay. That being said, if it doesn't provide your opponent with enough counterplay to cause serious problems, and if your king won't get checkmated, then it is good to use the king if you can.
Exploiting Enemy Weaknesses
It is a little self explanatory: try to give your opponent a difficult time with their weaknesses, using the principle of 2 weaknesses if you can. The tricky thing about this is about the sort of weaknesses your opponent will have. The obvious conclusion people often go into is that being up on material means is a big weakness, but in these endings, it is often the pawns that matter more than the pieces.
The main weaknesses in these endings are pieces being dominated or becoming super passive, your passed pawns, split pawns and (again) king safety. Again, the quality pawns matter more than the pieces (unless it is like a Queen vs Rook or something similar).
Passed pawns
In almost every single endgame, these are some of the most decisive factors to winning them. Passed pawns become especially important in these endgames. Passed pawns are very useful as they can tie down enemy pieces, allowing you to be more active. If you can, promote the pawn and get your opponent to sacrifice.
Being More Active
There are in fact 2 ways to be more active than your opponent. The first way is obvious: make your own pieces active. The second way is not as obvious but practically achieves exactly the same thing: restrict enemy pieces.
Activate your own pieces
I'm sure you've often wondered why pieces were assigned certain points. To make it brief, it is just the formula below:
(how many squares they control at most)/3 +- (unique feature)
For instance, a Bishop can control 13 squares at max, so 13/3 = 4.333... . However, as it is stuck to only one colour, it is denied from 1/2 the board, so is worth just over 3 points. But with another Bishop, the pair can control all the squares, so together are worth 7 or more.
A knight can control only 8 squares at max, so 8/3 = 2.666... . However, it's movement cannot be done by any other piece, so is roughly worth 3 points.
I could go on for other pieces, but you get the gist. To activate pieces in these endings, give them access to as much of the board as possible, focusing on the useful squares they control.
Restrict enemy pieces
If you can, simply attacking squares your opponent's pieces can go to significantly reduces their value. Even if it means that your pieces will have to control them for the time being, you will always get the first move when their piece can finally move. For instance, in the position below, white is bound to win easily, even while being up only 1 pawn (and that too, a Rook pawn!):
While that was not a piece imbalance ending, it shows how powerful restricting enemy pieces can be. Black was playing practically down a piece and a pawn.
Passed pawns can be really good at restricting pieces too:
Despite being down 4 points of material and having an equal number of pawns, white manages to win because of how well the passed pawn ties down black's Queen! A true demonstration of how piece values can change.
Also just simply cutting of squares physically is a really good way of making pieces useless:
How they may arise
The first thought that often comes to mind with how they appeared is that one side must have blundered early on, but very often they appear from sacrifices to try and make things difficult for your opponent or to try and win. One thing to note about these endings is that because they are super rare, there won't be any really simplified versions about these endgames written, especially since there are so many different piece imbalances that can occur. However, I will be doing game fragments to demonstrate them
In the first game below