An API is an Application Programming Interface. This is really a generic term for a lot of different sort of things. I'm assuming you mean "Web APIs" which are usually RESTful or SOAP based. The sort of API that is used really differs from one service to another. Some have multiple ways of working with the API, like Amazon has REST based and SOAP based services.
The best way to get started if you are doing this from the client side, is to really start learning Javascript and then learn JQuery. Then from there choose a service you'd like to program to. You could learn a lot of Javascript by coding for the Google Maps API. This is probably the best place to start as it is a pretty neat system, not too difficult, requires you to have a good level of knowledge of Javascript (which you can learn along the way).