There are two ways to acquire Twitter tweet data – historical and live “Firehose” feeds.
Historical Tweets via Sifter
Using Sifter you can search through the entire undeleted history of Twitter data from the inception of Twitter until the current time.
The cost includes licensing the data from Twitter plus a $20/day fee while using the service. Please see http://sifter.texifter.com/Home/Pricing for more information.
Gnip PowerTrack Livefeed
Twitter data can be collected more affordably using the real-time Gnip PowerTrack Twitter feed in DiscoverText, which is a day-forward version of the data stream from the time you start the data collection. This is also known as the "Firehose." Data begins to arrive as soon as you set up the data feed, and stops when you pause or delete the data feed.
The cost includes the cost of licensing the data from Twitter; there is no per-day fee.
More information about live feeds.