Voyant 2: Voyant 2 is a suite of tools for text analysis that can be used on either pasted text, documents from the web or uploaded files. You can also read the blog of one of its creators Stefan Sinclair here. Note that not all of the Voyant tools show up on the main interface. There is a great list of other Voyant tools here.
TAPoR 3: The Text Analysis Portal for Research (version 2), highlights Voyant as a tool but it has a set of curated lists for other resources. It is currently the best consolidation site for text analysis on the web.
Netvizz: Facebook has made it harder to do analysis than previously, but Netvizz is still useful
twXplorer: Similarly Twitter has made it harder to do analysis than previously, but twXplorer is still useful
Umigon: Umigon analyzes Twitter for "sentiments" (positive, negative, neutral) and is useful for looking at trends. It is probably the best opinion mining tool available for Twitter.
Netlytic: Netlytic works cross platform on social media (including Youtube, RSS, Instagram) and is good for data visualizaiton.
Overview: Overview is good for documents--analyzing lots of them.
Wordseer: Wordseer is an older text analysis project that relies on marked-up documents. Still useful for a number of things, but it requires your data to be structured in a certain way.
Mallet: Topic modeling analyzes clusters of words occurring together in order to refine or develop a topic for analysis. Mallet is one of the best packages for this.
Advanced Topic Modeling: David Blei's list of more advanced programs for topic modeling.
Updated 1-2017 Batchelor