Its a service offered by Amazon where you can hire people ("crowdsource") to do simple tasks, like post a review of a service or answer a quiz or whatever.

The name is based on a chess-playing 'computer' that was actually a man inside of a machine, who was outputting the machine's moves.

http://en.wikipedia.org/wiki/Amazon_Mechanical_Turk

Think of it as micro-jobs. It's a service that allows you to post small tasks for a small amount of money. The pay ranges from anywhere from a few pennies to a few dollars and takes anywhere from a few seconds to a few minutes to complete.

I believe the parent was suggesting you post the job requesting people signup for and install Dropbox. It's a flawed idea since it violates both service's TOS.

In fact, 7 of the 9 specific examples of prohibited activity [1] would fit this task

"HITs requiring disclosure of the Worker's identity or e-mail address, either directly or indirectly

HITs requiring registration at another website or group

HITs that directly or indirectly promote a site, service, or opinion

HITs that violate the terms and conditions of an activity or website (for instance asking Workers to vote for something)

HITs that generate "referred" site visits or click-through traffic

HITs that ask Workers to take action to manipulate a website's behavior or results

HITs that require Workers to download software"

[1] https://www.mturk.com/mturk/help?helpPage=policies

You pay people to perform small tasks: in this case, join Dropbox.