Simplest way to know that Kf3 is losing in this position?

     The important principle here is that black can draw only if he gets his king in front of the pawn while preventing the white king getting in front of the pawn. With Kg3 and Kh4 white gets around his pawn before black can get there. 

     The position white wants to reach is P on g4, K on g5, black K on g7 with black to move. Whichever way black then moves, white can advance his king the other direction, making it possible to escort the pawn to the eighth rank.

     This position is explained in every basic book on the endgame. Just studying basic endgame principles and knowing which are won, drawn and lost is an invaluable tool. It will enable you to decide whether to simplify into an ending or complicate play in the middlegame.

     Incidentally, you can doubtless figure out why this technique doesn't work with rook pawns.

Unfortunately not quite so simple. If you look at the position after two moves in my first diagram in post #14 viz:

It will take both kings 2 moves to get to h6 (or f6, g6), but white moves first. 

That doesn't help. He still draws.

If it were Black to move, he should not approach the key squares. Only 1...Kg8 draws (to reach e.g. @llama47's third position in post #17).

Somebody sent me a more interesting position similar to the above but with the kings more remote from the key squares:

"I do not think that IMBacon is explaining it properly. Consider this position: 

It will take both kings 3 moves to get to f4 and white moves first, but the result is nevertheless a draw."

(So someone agrees with me.)

He also pointed out that @IMBacon's references to opposition in the accompanying diagram are suspect.

Here, IMBacon's comment, "2.Kh4 White takes the distant opposition", is wrong; it's not opposition. (Also most people would not refer to the position after Black's 2...Kf6 in the first variation as direct opposition. That's just nomenclature, but since it does Black no good it's difficult to understand why he talks about opposition at all at that point.)

If you're into rules, the following might be useful for non rook's pawns:

(A)  If the White king reaches the side of his pawn, Black's only possible drawing moves are to either 2 or 4 squares ahead of one of the White pieces on it's file. (None, if the White pieces are on the 7th. rank and only opposition on the kings file if the White pieces are on the 6th rank.)

(B) If the White king reaches one of the three squares immediately ahead of his pawn Black's only possible drawing move is to stand in direct opposition ahead of the king on it's file, but this defence disappears if the pawn has crossed the crease.

There is an exception to (A) namely the position below and its counterpart on the king's side. Black also draws if he can move to the corner (leaving the position  shown) in this case.

There is an exception to (B) namely the position below and its counterpart on the king's side, when Black doesn't have to  (indeed can't ) move  at all.

It would probably be useful for you to check these rules are correct.

E.g. here, by (A) Black to move must play Kd6 - any other move loses..

If the Black king were instead on e5 he would have a choice of d4, e4, d6 or e6. If he were on his starting rank, for example,  he'd be lost.

In the position you give it takes two moves for the White king to reach either side of his pawn so you just need to check the number of moves the Black king needs to reach the squares f6/8, g6/8 and h6/8 to decide which side you need. (Note g6/8 would be good enough for either from Black's point of view.)

As an example of (B), here Black to move must play Kd5, which means he's lost unless his king is on one of the squares c5, c6-e6 or e5.

My correspondent gives a rather full analysis of your position based on (B) which I'll reproduce shortly.

As for rook's pawns Black just needs to block the rook's file ahead of of the pawn with one of the kings (or take the pawn). I'll expand on that shortly also.

Edit: Note I added a necessary amendment and counterexample to rule (A) for 6th. rank pawns.

Wow king-pawn endgames are complicated.

I have a newbie-ish type question on the position in the OP

Is the problem here that white is to move and he is already in Opposition?  I.e., I'm looking that the "rectangle" that the two kings make, and all four corners are dark squares.

In other words, if it were black's turn to move, then white just needs to keep the opposition and he wins, right?

Do I understand this position/strategy correctly?

Black has the opposition, but the opposition is not important until the White king has advanced further than the pawn when Black must be able to reply specifically with direct opposition along the file according to rule (B) I gave in post #23. Before that, if the White king draws level with the pawn then Black has two files on which to maintain the draw according to rule (A) in the same post and doesn't need opposition. 

That assumes he draws level adjacent to the pawn or emerges on one of the squares in front of the pawn. Black has the threat of reaching the critical square g5 in 3 moves which limits White's possibilities in that regard. If Black makes it there White's win is goneski wherever his king is. White can only prevent it by reaching f3 or h3 in at most 3 moves assuming he doesn't advance his pawn, but emerging on e5 would necessarily have let the Black king to g5, so we can assume the White king reaches f3 or h3 at some stage (if the pawn doesn't advance).

At that point according to rule (A) Black draws if and only if he can reach one of f6/8 or g6/8 if it's f3 or if he can reach one of g6/8 or h68 if it's h3.

In fact if White moves directly to h3  in two moves Black cannot even make the g file in the same two moves so he loses. But if White moves to f3 in two moves Black can make either f6 or f8 and it's a draw.

But that is a matter of legs not opposition. Opposition is a red herring. 

Advancing the pawn at any stage effectively just gives Black extra moves. He would then draw even if White advanced his king on the h file. E.g.

White to move wins, black to move draws.  White needs to get his king in front of the pawn with opposition.  Example:  1.Kg3 Ke7 2.Kh4 Kf6 3.Kh5 Kg7 4.Kg5 Kh7 5.Kf6.   Or, 3...Kf7 4.Kh6.   Or, 2....Kf7 Kh5 3.Kg7 Kg5 etc.  With black on the move, 1...Ke7 2.Kg3 Kf6 3.Kf4 Kg6 4.g5 Kg7 5.Kf5 Kf7 6.g6+ Kg7 7.Kg5 Kg8 (not Kf8) 8.Kf6 Kf8 and white can't make progress.  Note that it is important is to move the king straight back on move 7, otherwise  7...Kf8 8.Kf6 Kg8 9.g7 and white wins.

@Knights_of_Doom

In your first variation (starting, "Or, 3...Kf7") in the White to play example White reaches the fifth rank without the opposition and remains without the opposition - but is still in a winning position. Why do you say, "White needs to get his king in front of the pawn with opposition"? 

Obviously my rule (B) in post #23 would imply that is a sufficient condition for White to win, but it isn't a necessary condition. 

In fact if the White king can play to one of the three squares in front of the pawn and Black can't immediately reply with close opposition ahead on the same file, that is enough. After 3.Kh5 Black is simply on the wrong file.

Be a supercomputer

Herewith the analysis I was sent based on rule (B) (as promised):

"In this example

(White king on f2, White pawn on g4, Black king on d8, White to move),

it somewhat muddies the water that Black is able to prevent the White king reaching f5 or g5 by the most direct route, but Black is unable to prevent the White king moving directly to h5. It would be a somewhat cleaner illustration of the basic idea if the position started like this:

(White king on f1, White pawn on g3, Black king on d8, White to move).

Now:

If White is able to play Kf4, Black can only achieve a draw by replying with ...Kf6.

If White is able to play Kg4, Black can only achieve a draw by replying with ...Kg6.

If White is able to play Kh4, Black can only achieve a draw by replying with ...Kh6.

If White goes for f4, then Black can get to f6 with time to spare:

1 Kf2 Ke7 2 Kf3 Kf7 3 Kf4 Kf6 1/2-1/2

If White goes for g4, then Black can get to g6 with no time to spare:

1 Kf2 Ke7 2 Kf3 Kf7 3 Kg4 Kg6 1/2-1/2

If White goes for h4, then Black cannot get to h6 in time:

1 Kg2 Ke7 2 Kh3 Kf6 3 Kh4 1-0

I should illustrate two important exceptions to the basic idea.

A. If the pawn is on e5 and the White king is on d6, then Black will not be able to draw by moving to d8 (or anywhere else, for that matter).

(White king on d6, White pawn on e5, Black king on e8, Black to move)

1...Kd8 2 e6 Ke8 3 e7 Kf7 4 Kd7 1-0

B. If the pawn is a rook pawn, then a wide variety of positions enable Black to achieve a draw.

(White king on g5, White pawn on h3, Black king on e8, White to move)

1 Kg6 Kf8 2 Kh7 Kf7 3 h4 Kf8 4 h5 Kf7 5 h6 Kf8 6 Kg6 Kg8 7 h7+ Kh8 8 Kh6 1/2-1/2"

A check is usually a good omen is what i do.

It would have been more accurate for me to say:  "white needs to get his king in front of the pawn without the opponent getting opposition".   Further, I probably should add that if the king is two or more squares in front of the pawn, it doesn't matter who has opposition (although that doesn't apply in this position).

@Knights_of_Doom

Strictly accurate would be, "white needs to get his king in front of the pawn without the opponent immediately getting direct opposition ahead on the same file" (my rule (B) in post #23). In OP's position the opponent can't make it to the h file (with or without direct opposition).

Two squares in front of the pawn fails in the general case if the opponent's king has already reached a critical square of the pawn and three squares in front fails if the opponent's king has reached a critical square or can reach one on the next move. More than three squares increases the number of exceptions. (Critical squares in this case are the squares directly to either side of the pawn or directly in front. In the case of an attack from behind the pawn just runs home; if his own king blocks the path this doesn't alter anything.)

Although in post #9 I dubbed the squares three moves ahead of a pawn on it's starting square "key squares", there is one exception additional to those for the key squares two moves ahead namely this position (and its reflection in the pawn's file).

But an attack from the critical squares to the side of the pawn fails in this case because of the pawn's double first move .

That position moved up a rank is drawn.

Obviously, for the key squares one rank in front of the pawn (pawn past the crease) an attack on the pawn is not an exception. The only exception is the position I showed as an exception to rule (B) in post #23.

(Of course none of that takes rook's pawns into account.)