From its starting square, a knight has 3 legal moves: The “standard” one (c3, f3, c6, f6), the “weird” one (a3, h3, a6, h6) and the “other” one (d2, e2, d7, e7).
In my games, I almost always do the standard move. And I completely overlook the other move by my opponent, to my detriment. When are the weird and other moves preferred?
It's probably first worth revisiting why the "standard" moves are normally preferred.
A white knight on c3 defends white's central square e4 and attacks black's central square d5. That is that knight attacks/defends/supports 2 out of the 4 central squares. In the opening those 4 squares are the most important squares on the board (unless you fall for a cheapo).
In contrast a knight on d2 only defends white's central e4 square. On d2 the knight is only half as effective as it is on c3. By symmetry the same arguments apply to knights on f3, c6 and f6.
In general in open games these four squares are the best ones for the knights on their first move. In closed positions it is not quite so clear cut. Often other factors make other squares better or just make those squares impossible.
In closed and semi-open openings these squares may not all be available and moving a knight there may not be possible. The Advance Caro Kann is probably the best example of this. After:
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1. e4 c6 2. d4 d5 3. e5
black can't put a knight on c6 because it is already occupied by a black pawn and f6 is also out of bounds because it is attacked by the white pawn on e5.
In the Winawer variation of the French Defence white puts a knight on c3 but this is a very committal move. Black can pin the knight with Bb4 and a double edged tactical battle is likely. Consequently at higher levels Nd2 is preferred heading into a Tarrasch variation of the French. This satisfies the key requirement of the position, defend the attacked white pawn on e4, while also avoiding the pin from the dark squared black bishop and allowing the d4 pawn to be defended with the move c3.
The other reason for moving a knight to d2/7 is to protect the other knight developed on f3/6 when it is pinned to the queen thereby allowing the queen to move without risking doubled pawns if black decides to exchange bishop for knight.
Finally moves like Nh6 use the h6 square as a staging post aiming for f7 or f5. These ideas come up regularly in the French Defence. Here are a couple of examples:
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1. e4 e6 2. d4 d5 3. e5 c5 4. c3 Nc6 5. Nf3 Bd7 6. a3 f6 7. Bd3 fxe5 (7...cxd4 8. cxd4 Nh6 {with the idea of Nf7 to put more pressure on e5 and if} 8. Bxh6 gxh6 9. Nh4?! Qb6 10. Bxh6 gxh6 11. Qh5+ Kd8) 8. dxe5 Nh6 9. O-O Nf7
If you play openings like the French or the Caro Kann as black then you will become more familiar with these ideas. In any case you should be mindful of possible advantages in playing Nd2/7 to protect a pinned knight on f3/6 making it easier to break the pin and possible advantages in leaving c3/6 free for a pawn to support a pawn on d4/5 or to kick a bishop on b4/5.
This question is too general to have a definitive answer. There is no universal criteria to know where to develop a knight. For instance, after 1.e4 e6 2.d4 d5 3.Nd2 dxe4 4.Nxe4, now 4...Nc6 would be the "weird" move, as it would make it harder for Black to push ...c5 which is one of his main plans in this position. 4...Nd7 is therefore a much more natural move.
Sometimes the "normal" square is simply not available. Think of 1.e4 c6 2.d4 d5 3. e5
Another interesting example is 1. d4 f5 2. g3 Nf6 3.Bg2 e6 4.c4 d5. Now 5.Nh3 is an interesting way to reach the f4 square, even though 5.Nf3 targetting e5 is probably the main option.
Finally, keep in mind that the first square we move our knight to is probably not its "destine". After 1.e4 c5 2.Nf3 d6 3.d4 cxd4 4.Nxd4 e5 5.Nb3 Nc6 we develop our b1 knight to c3 because we dream of putting it on d5. If its best square happened to be on c4, Nc3 would suddenly become the third best option.
Much more likely it is your other moves that cause your problems not that you did a 'standard' horsie placement and they used one that was not.
The "standard" movement of the knight in the opening is pretty much forced. Other than the Alekhine's knight movement (which is also inferior but not busted), other options of knight movement almost always give >1 advantage to the other side, busted and convertible by engines.
