When you give the player an ability or boost or otherwise some form of mechanic, it will likely provide at least one of these four things (but likely more):
Toy represents all things good and fun. Whether it’s the amusement gained through seeing yourself through another’s eyes (Clairvoyance, from Psychonauts), or the sheer joy of gently nudging your opponent to a humiliating death (Prod, from the Worms series), this aspect of a game mechanic is to make the experience fun. There are certainly more involved mechanics for which this element is an important part; in Cave Story, none of the myriad of weapons is necessary — but through their variety they provide the player with fun. You make the choice: do you want to destroy your enemies with bubbles? Rolling balls of fire? Or maybe with ducks? Speaking of choice…
Fork represents decision-making. Not choices like “do you want a green hat or a red hat?” as, assuming these things are entirely superficial, they would fall under the category of Toys. On the other hand, a Fork allows you to make a choice that affects the game. Choosing your race or class (The choices provided by Nethack are a prime example) can affect available skills, starting stats, and the progress of the game as a whole. Gaining three different spells — ice that freezes, fire that burns over time, and lightning that jumps between enemies — grants you new choices. Upon encountering four enemies, one of whom is a leader, what do you do? The lightning spell might deal the most damage overall, but the ice spell could disable the leader (or a minion) while fire could more quickly dispatch of him. Another example might be Bombs (from Spelunky) — these bombs are highly useful (especially once sticky) but you have a limited supply. It’s up to you to choose
Key represents a requirement. At its most basic, an example of a Key would be a key for an unbreakable and perfect door. There is no Fork, and there is little Toy in such a thing. However, that is not to say there are Keys without fun or choice — remember I’m talking about a ‘pure’ Key. As an example, Bombs (from various Legend of Zelda games) are a common Key. If there is a wall or rock that looks cracked, you blow it up. From the same game, there are Silver/Golden gauntlets who allow you to lift larger rocks than normally possible. These things are Keys, although they are utilized in puzzles which require a bit of thinking.
Trial represents the testing of a player’s skills. Almost every Trial will have a reward for success, and a punishment for failure. While games frequently are Trials in and of themselves, a Trial within an ability means one needs to use the ability correctly. In most rhythm games your abilities are restricted to sticking to the rhythm; these abilities are Trials, as you simply succeed or fail. Every command in Patapon is a Trial, but it’s a rhythm game too. Shooting at moving targets in an FPS? Trial.
I’ll get into this a bit deeper later.