Twitter's data is massive - 1 week is the time it takes users to send a billion tweets. Currently it isn't possible to search older tweets comprehensively for free. Twitter provides search access to tweets no more than 10 days ago (or scroll back to the most recent 3200 posts); other search engines may retrieve only "popular" tweets beyond 10 days ago.
In April 2010, Twitter donated its entire archive of public tweets to the Library of Congress but this archive is not yet available for researchers. (The Library of Congress Blog posts updates to their progress. January 2013 update.)
In December 2012, Twitter released a tool for users to download their entire tweet history (only for your own tweets, not others' tweets).
"Our Social-media Amnesia" by Nancy Scola for Reuters (March 21, 2012) is a good overview of the issues of accessing the full archive of Twitter.
This is a list of search engines to find recent Twitter posts. See below for strategies on finding older tweets.
Strategies to find older twitter posts (for free*):
Google retrieves less older tweets since its deal to access the full Twitter data stream ended in 2011. Use the following search syntax to compare results with other search engines:
e.g. site:twitter.com #jan25
*Twitter rents access to its data "firehose" (all public tweets) to companies, e.g. Gnip and DataSift. These companies offer their services for a fee.
A quick, easy, and free citation generator that converts Tweets into, properly formatted MLA and APA, citations. Paste a link to the Tweet you want to cite in the field below, click 'Go', and you're on your way!
MLA Citation Style: see How do I cite a tweet?
Ibrahim, Gigi (gsquare86). "Army tanks have arrived by us embassy http://twitpic.com/8u0pjv.” 9 March 2012, 10:30 a.m. Tweet.
APA Citation Style: see APA Quick Answers
Ibrahim, Gigi. (2012, March 9). Army tanks have arrived by us embassy http://twitpic.com/8u0pjv [Twitter post]. Retrieved from http://twitter.com/#!/Gsquare86/status/178140680042975233