Book subject: Teaching you the system

Remember, game companies want you to have fun.  It is in their best interest that your experience be as enjoyable as possible.  Because video games are an interactive media, you're actually involved in that experience.  It is therefore vital that the developers know you understand how to use the system, or at least how to get the rewards they most want you to get.  This can sometimes be a challenging task.  The Xbox 360 alone has 4 main face buttons, 4 buttons for your index fingers on both hands, 2 center buttons used more for minor functions, and even the 2 analog sticks can be pressed for further functionality.  Sometimes games will use both analog sticks while incorporating the D-Pad, essentially making 4-8 more buttons.  That's as much as 20 functions (plus sticks) to manage on a controller!  This is obviously an extreme example of how complicated controllers can get (run a search for Steel Battalion if you really wanna see crazy), but even using 1 stick with 4 buttons (face or index) is similar in difficulty to driving a contemporary vehicle.

To illustrate a prime example of how companies teach mechanics to players, I want to turn to Valve Software.  They're constantly coming up with new mechanics for players to experience.  Their game Half-Life 2 is generally a first-person shooter.  For the uninitiated, it simply means you see the game literally from the protagonist's perspective and you basically shoot everything that moves (there's more to it, but that's what grabs you).  Throughout the game, they have you operate a bunch of different devices that are a whole different world from aiming at stuff to destroy.  The holy grail of this is the Gravity Gun.  It's just about the only weapon you get that's not inherently destructive.  Instead of shooting things, it's grabs stuff and chucks it around.  Unlike other area-specific mechanics, you carry this one with you everywhere.  It even plays a dominant role in Episode 2.
To help you fully understand this equipment, they go through a 3-layer system.  They first show you how it works without being actively involved in it.  Episode 2 accomplishes this by having your partner grab a sealed door with it to get you out of a crashed train car.  You can't even get to her and everything around you directs your attention to this door.  You can only avoid seeing this action by trying (or you're honestly just that interested in examining every little bit of the graphical design).  When she hands it to you, the game tells you what buttons do what function and you have to use it to navigate around for a while.  You don't encounter any threats at this point.  They wouldn't want to hinder the experience by making you fight hostiles while you fight your own confusion.  They ease you in a bit more by offering a small amount of enemies in areas filled with highly explosive canisters, or at least a bunch of stuff you can throw at them.  This process is especially exaggerated in Half-Life 2 (not Episode 2, different game), where an entire section of the game is designed to show you all the cool things you can do with the Gravity Gun.

If all that got you lost and confused, don't worry.  Plenty of games have been teaching you things in far more basic ways.  Nintendo's always known this idea.  In the above screenshot of Metroid, you normally approach this room from the upper-right corner.  Stairs basically get you up there and you drop down to collect your first upgrade: the ability to turn into a small ball.  The ledge is now too high for you to reach by jumping.  Once you figure out how to change (simply pressing down on the D-Pad), it should easily occur to you that you can fit through the small tunnel on the bottom.  Legend of Zelda games have always taken this approach ad nauseum.  Once you get the special tool for your level/dungeon, they make a great many situations to show you all the different things you can do with it.

This was made easier from Ocarina of Time onward.  Your fairy companion, Navi, was always with you.  You could always talk to her for assistance.  If you were taking too long to accomplish a task, she'll even freely offer up "suggestions" on how to better approach the problem.  These "suggestions" are of course Nintendo directly trying to teach you how to progress while trying to plausibly keep you absorbed in the game world.  This approach is a mixed bag.  On the plus side, having something directly tell you what you should be doing all but ensures you don't get stuck in one specific area.  Getting stuck is a big reason people will put down the controller and never try again.  If you know how to play through the game, you can get more of the game's story and cinematic payoffs.  However, this is a very tricky thing to do right.  Get too explicit, and it can feel like the game is playing itself for you.  The challenge of solving puzzles is also sometimes the entire appeal of a game.  Get too simplistic and it's like you've done nothing at all.  Do it too much, it will be irritating.  Legend of Zelda and Star Fox already have those reputations well set with such phrases as "Hey! Listen!" and "Do a barrel roll!" that simply appear way too much.  It gets as irritating as "are we there yet?"

How can you use this to your advantage?  Of course the simplest answer is just to pay attention, dummy!  I know, if you were even asking the question you already feel you've been doing that.  If help is given through text, it will often have certain words in bold or a different color.  If not, they could still be using words or phrases that stand out.  They can feel forced in unnaturally.  These words will often match or at least relate to a word in something else.  Check your map or documents to see anything with such words or phrases, even if they don't initially seem connected.  Of course, all that is related to puzzle-solving.
What about action games trying to teach you a mechanic?  If you're stuck in someplace where something new was introduced and your character is dying a lot, patience really is the key.  Look for similar situations that are not so dangerous and use that environment to experiment.  If you have no choice but to learn on the fly in a dangerous environment, remember nothing is actually on the line when you play a video game.  Take it for granted that you will be killed a lot and don't even pay any attention to the threats more than you must.  Your primary objective is finding out what you have to do and how to do it.  Only attempt to execute it when you're confident you know this.  Think of the hordes of enemies coming at you merely as a time limit you have before you have to restart the session.  Worst-case scenario, go outside the game (the developers failed to teach you properly) and seek assistance elsewhere.  We all had to do it with the original Legend of Zelda; either by brainstorming with friends or buying guides.  That was back before we had the internet to just give us the answers and we had to trek 8 miles in the snow uphill both ways to find a store selling the guides we needed.