How the NSA actually gets the data depends on the type of information requested. If the analyst wants someone's private emails or social media posts, the NSA must request that specific data from companies such as Google and Facebook. For information that is already flowing through Internet cables that the NSA is monitoring, or the audio of phone calls, a targeting request instructs automatic systems to watch for the communications of a specific person and save them.
It's important to note that the NSA probably has information about you even if you aren't on this target list. If you have previously communicated with someone who has been targeted, then the NSA already has the content of any emails, instant messages, phone calls, etc. you exchanged with the targeted person. Also, your data is likely in bulk records such as phone metadata and internet traffic recordings. This is what makes these programs "mass surveillance," as opposed to traditional wiretaps, which are authorized by individual, specific court orders.
Even without the content of all your conversations and text messages, so-called "metadata" can reveal a tremendous amount about you. If they have your metadata, the NSA would have a record of your entire address book, or at least every person you've called in the last several years. They can guess who you are close to by how often you call someone, and when. By correlating the information from multiple people, they can do sophisticated "network analysis" of communities of many different kinds, personal or professional -- or criminal.
Phone company call records reveal where you were at the time that a call was made, because they include the identifier of the radio tower that transmitted the call to you. The government has denied that it collects this information, but former NSA employee Thomas Drake said they do. For a sense of just how powerful location data can be, see this visualization following a German politician everywhere he goes for months, based on his cellphone's location information.
The type of data can be used to discover the structure of groups planning terrorism. But metadata is a sensitive topic because there is great potential for abuse. While no one has claimed the NSA is doing this, it would be possible to use metadata to algorithmically identify, with some accuracy, members of other types of groups like the Tea Party or Occupy Wall Street, gun owners, undocumented immigrants, etc. An expert in network analysis could start with all of the calls made from the time and place of a protest, and trace the networks of associations out from there.