Twitter API refers to two specific Application Programming Interfaces (APIs) that can be used to download tweets in order to perform big data analysis. These two APIs for retrieving tweets from Twitter are public. The first API is a RESTful API, but as this API is synchronous it is also extremely limited in the number of retrieve calls it can work with. More information about the RESTful API is available here.
The Second API is a Streaming API providing a limited stream with roughly fifty tweets per second via a persistent HTTP connection. It is important to understand that only 1% of the real emitted Tweets of Twitter in every given point in time are available for free as download via the Streaming API. There are however two different public streams served via this API. The first filtered stream offers restrictions in order to retrieve tweets with configured filter options. While this is useful for geolocation and keywords in general, it is not possible to set a filter for certain languages. The second stream-based API option is a sample stream offering a random selection of tweets. More information about the Streaming API is available here.
Finally, the a above two APIs are providing a very limited access to the costly Twitter Firehose where more data can be obtained.
Twitter API Details
The following interesting video provides more pieces of information about this topic: