Twitter-API

The  Twitter API  is a proprietary API for accessing Twitter content and services.

Twitter for iOS
Twitter for iOS (in Apple's App Store) is an official "native" client app provided by Twitter.com

Custom API root
The Twitter iOS client has an option to use a different API root, "intended for a Twitter proxy server".

It may be theoretically possible to re-implement enough of the Twitter API on your own site to set your own site as a "Twitter proxy server" and thus use the Twitter client as a client to post directly to your own site (and perhaps read from it as well).

iPhone / iPod touch
5 steps to change your API endpoint root.

http://farm3.staticflickr.com/2857/9320856337_57271cbe41.jpg
 * 1) Tap on the "Me" tab

http://farm6.staticflickr.com/5453/9323649762_50b25a2770.jpg
 * 1) Tap on the "account switter" button

http://farm6.staticflickr.com/5333/9323649914_2b9f9f3fd7.jpg
 * 1) Tap on the "+" to add a new account

http://farm4.staticflickr.com/3785/9323650056_16c6148257.jpg
 * 1) Tap on the gear icon to change the new account's settings

http://farm4.staticflickr.com/3703/9323650178_850593ec1d.jpg
 * 1) Enter an API endpoint other than the default

iPad
5 steps to change your API endpoint root.

http://farm4.staticflickr.com/3707/9320857341_7b6039e67b.jpg
 * 1) Tap on the "Me" tab

http://farm8.staticflickr.com/7282/9323650640_7d4cdd2969.jpg
 * 1) Tap on the "account switter" button

http://farm4.staticflickr.com/3757/9320857639_7760131577.jpg
 * 1) Tap on the "+" to add a new account

http://farm8.staticflickr.com/7334/9323650992_fbe26efbc7.jpg
 * 1) Tap on the gear icon to change the new account's settings

http://farm3.staticflickr.com/3804/9320857959_6cd008601e.jpg
 * 1) Enter an API endpoint other than the default

Twin (Ruby)
Twin is a Ruby library that was extracted out of a client job by Mislav Marohnić. Twin implements a "subset of Twitter's API".

WordPress.com
http://en.support.wordpress.com/twitter-api/

More research is needed on the current status of this.

Tumblr.com
http://staff.tumblr.com/post/287703110/api

More research is needed on the current status of this.