In order to make authorized calls to Twitter’s APIs, an application must first obtain an OAuth access token on behalf of a Twitter user (or, issue Application-only authenticated requests, when user context is not required). The way to obtain such tokens will depend on the use case.

If you... Use...
Want to offer a “Sign in with Twitter” button on your website... Sign in with Twitter
Want to read or post Twitter data on behalf of visitors to your website... 3-legged OAuth
Have a mobile, desktop, or embedded app which can’t access a browser... PIN-based OAuth
Just want to access the API from your own account... Tokens from dev.twitter.com
Offer an API where clients send you data on behalf of Twitter users... OAuth Echo
Want to issue authenticated requests on behalf of the application itself Application-only authentication

Once you have an access token and token secret, the Twitter API is your oyster! By following the steps described in Authorizing a request, you can issue authorized requests to the REST API and the The Streaming APIs.

If the OAuth process sounds like it is beyond the scope of your integration, consider using Web Intents, which do not need to use access tokens to interact with the Twitter API.