In Chess, are there simple "mechanics" that amateurs can learn to improve their game? In the board game Go, for instance, it is often of great importance whether or not a stone can be captured in a so-called "ladder" (an often complicated formation). One of the things that separates professionals from beginning and intermediate players is that professionals can solve with a glance a problem that others struggle with.

Are there similar "mechanics" in chess (most likely the end game), that amateurs would do well to learn to greatly improve their game? Examples might include the ability of a king, or other piece to "catch" a passed pawn; the ability of a king to gain the opposition against the opposing king halfway across the board; the minimum number of moves needed for a bishop or knight to reach a certain critical square, etc.
Good question, and I think that there are a lot of different common patterns/tactics that improving players would do well to learn:

(very roughly ordered from simplest to most difficult)

"Simple" tactics and endgames


knight forks and bishop forks - get in the habit of just seeing the squares that would be forked instead of needing to spend time looking
rook and queen forks - very similar to the above, but sometimes these are more difficult to see for some reason (i.e. white king on g1, knight on g3, Rg2+ would fork the two if the rook is defended)
king and pawn endgames - as mentioned in your question, the rule of square is crucial.  At the end of a long variation leading into a king and pawn endgame, just knowing whether or not the pawn will queen is often the difference between a win and a loss, and this trick will save a lot of time, often during time pressure.
drawn bishop and rook pawn endgames - again, if you know that a rook pawn and the wrong colored bishop will only draw, this will enable you to save time and half points.
drawn knight vs pawn endgames - sometimes a lone knight can fight successfully against the enemy king and pawn, sometimes not.  At a certain level (1900+?) it's important to know when the king and pawn win and when they only draw.  That's probably a separate question on its own...


Rook endgames


Lucena position - the Lucena position is the most common method of winning with a rook and extra pawn vs rook.  It's not terribly difficult to commit to memory either - spend 5-10 minutes and learn how to win when you get your king in front of your pawn!
Philidor position - the Philidor position is the most common drawing method if you are down a pawn.  Again, doesn't take much time to learn the basic position, and a good investment if you want to save a few draws in your games.
cut off king - if the enemy king can be cut off along a file or a rank, usually the stronger side will win.  However, learning the exceptions to this rule is important as well.  This is far less common than other winning rook endgames though, so less important if you just want practical knowledge.  


Piece relationships and domination


bishop dominates knight - a picture is worth 1000 words...

king dominates knight - similar to king and pawn vs lone knight...



More complicated endgames


queen endgames - learning how to win with an extra pawn in a queen endgame can be valuable, as can learning how to defend.  There are few very common tricks (walking your king close to the enemy king to allow for cross-checks, deciding which corner to run to when you are on the weaker side, etc.) that are useful.  Again, this might be worth its own question.
bishops of opposite color - many times bishops of opposite color are a quick draw.  Not always though!  Learning some of the patterns used to get a win with an extra pawn or two in this endgame will allow you to win as the stronger side, and avoid losing as the weaker side.  Common patterns/tricks include two pawns vs no pawns separated by a single file (many times a draw), learning how to win with two pawns separated by 3+ files (hint: king helps the pawn barricaded by the enemy bishop), and learning how to draw these positions.


Tactics and mating patterns

Learning some basic mating patterns or typical tricks can be useful as well:


Greek gift Bxh7+ sacrifice - if black doesn't have a knight on f6 and white has a knight on f3, queen on d1, and a bishop that is attacking h7, it's imperative to at least check this sacrifice.  The Art of the Attack has a whole chapter dedicated to this common theme.  
Checkmate patterns - spend an hour learning patterns - the wikipedia article is a great start, but Art of the Checkmate is the authoritative book and it is available fairly cheap on Amazon...


Well, that was a lot, please feel free to add or embellish this answer if you think of more patterns and tricks!