You can think of API testing as a mix of different strategies, each having its own purpose. Unit testing is all about making sure that your code consumes the API in a reliable way. Functional, or black box testing, lets you know how the API replies to specific requests. Load testing follows the functional testing patterns making repeated API calls, measuring response times.
Are there any tools that help you manage your API testing? Find more information at this presentation on How to Automate API Testing.