Author(s):

Chris Messina

In 140 characters:

via is used to credit one or more Twitter users as a source.

Description:

Unlike retweeting, via is used to cite a source when the content of the tweet is largely original and not simply "forwarded". 

Details and Use Cases:

To use via, one simply needs to insert "via @username" or "/via: @username" into a tweet:

VIA: Alternate (to discuss)

Author(s):

Sanjay Kairam

In 140 characters:

via is used to cite a source for referred URL without referencing original content of the tweet.

Description:

As opposed to Chris' observed usage (and pointing to the large diversity of usage!), I use and often see via as a way of citing someone as the provider of a URL or piece of information without implying that you are directly quoting them (often useful when you need to modify the original tweet due to character limitations).

Details and Use Cases:

Same as above: To use via, one simply needs to insert "via @username" or "/via: @username" into a tweet.

Author(s):

Sachin Agarwal

In 140 characters:

/v or ~ is used to cite a source for referred URL without referencing original content of the tweet.

Description:

To save characters (two and three, respectively), /v @username or ~@username could be substituted.  ~@username, in particular, is well suited for the beginning of tweets.

Details and Use Cases:

Same as above: To use via, one simply needs to insert "/v @username" or "~@username" into a tweet.

Criticism

Colin Devroe points out that via may be subsumed by the new built-in retweet functionality.

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