Ever since the Xbox 360 was released gamers have been collecting
GamerPoints for every game they play. GamerPoints are awarded for
completing a certain task or Achievement, in a game. This can be
as simple as pressing start (The Simpsons Game, 5G), or as
tedious as killing 100,000 enemies (Gears of War 2, 50G). The
amount of GamerPoints earned usually depends on the difficulty of
the Achievement. Retail games usually have a total of 1000G and
Xbox Live Arcade Games have a total of 200G, though both these
totals can be increased by purchasing downloadable content.
GamerPoints are then stored on the players Xbox Live account, and
GamerTag, and displayed on their GamerCard. For example my
GamerTag, Lardy136, currently has 33529 GamerPoints.
This was a brilliant move by Microsoft, so brilliant that the Sony Playstation 3 released their own form of Achievements, called Trophies. The concept is basically the same but instead of collecting GamerPoints players collect Trophies. There are 4 types of trophy Bronze, Silver, Gold and Platinum. The trophy earned depends on the difficulty of the achievement; Platinum trophies are awarded when the player has obtained every other trophy in a game. Each Playstation Network player has a trophy level and a percentage bar that increases with each trophy gained. Platinum Trophies are worth more than Gold Trophies, Gold Trophies are worth more than Silver... etc.
The main purpose of GamerScore and Trophies is bragging, players can compare with each other to see who's earned the most achievements, or maybe just a particular one that's very difficult. Achievements also encourage players to replay games as most games require several playthroughs to earn every achievement. Some are hidden until they are earned, encouraging the user to explore the game(or use a search engine) to find out what they are. Achivements may even unlock things in other games, for example if you have unlocked the "A Dish Best Served Cold" Achievement in Gears of War then General Raam is unlocked as a multiplayer character in Gears of War 2.
There is one major downside; players are so used to having achivements as a reward that they are hesitant to play retro games or games on other consoles that don't have them.While this isn't a problem for Microsoft or Sony, it's a problem for Nintendo. But since Nintendo consoles are usually aimed towards the young or the occasional granny it's not a major concern, until they decide to draw in teenagers and young adults who are the majority of serious gamers. The answer to this is to put the older games onto the Xbox Live Arcade (or PSN) and add achievements; this has already been done to some games but not enough in my opinion.
So do you get satisfaction from earning achievements? Is it worth neglecting older games with no achievements to brag about? You decide.