What exactly is API in EVE? Where can I get one and how does it work? How does it help me use third party apps?
Eric Dahlgren
[ Editor ]
The EVE API is a service provided by CCP that supplies account and character information to external applications without requiring you to log into the game. API Keys are used by applications and websites to display currently training skills, character sheets, wallet balances, and market and corporation information. The API can only be used to retrieve information, and does not allow external applications to update your skill queue or make transactions without logging into the game.
There are two types of API Keys: Limited Access and Full Access. A Limited Access API Key can retrieve Skill Training and Character Sheet information only. This is used by skill monitors like EVEMon to track character progress, and is requested by some corporations when applying to view your Character Sheet and a list of characters on your account.
The Full Access API Key can retrieve all of your character information including wallet transactions, market orders, and corporation information. Because the Full Access API Key provides so much information about your characters, it is recommended that you only supply it to individuals and applications you trust.
You can retrieve your Limited Access API Key and Full Access API Key or generate a new keys by visiting the EVE API Key Management page of the EVE Online website. You can also view your API Access Log.
Always better to summarize the difference here than referring out to a website (something I am guilty of here and there too).
I have modified my answer to be more complete without linking off-site in the spirit of making STC the most complete EVE Knowledge-base. :P I hope nobody minds.

This question has already been asked, as posed, though it contains the seed for a new line of questioning (“third party apps” focus). This question should should be merged with the formerly asked question (http://skilltrainingcomplete.com/questions/600/api-key-what-is-it-where-can-i-find-it).
Merged as suggested