A Takedown Notice orders the removal of one or more files that have been shared in apparent violation of copyright. This type of notice is sent directly to MIT and typically contains the IP address of the machine involved, the time and date of the file transfer, the name and size of the file, and what the file is believed to contain (e.g. a specific song or TV show). There is no distinction between whether the file transfer was an upload or a download. This notice does not require MIT to identify you, just that it ensure that you have removed the file.
A Subpoena Notification is a written notice (including a copy of the subpoena) originating from MIT's Senior Counsel's office informing you that a subpoena has been validly served on MIT in connection with a lawsuit filed concerning a certain IP address that is registered to you. This means that a lawyer acting on behalf of a copyright holder has asked the Court to compel MIT (not itself a party to the lawsuit) to release identifying information (i.e., yours) relating to the IP address at the particular time and date that the infringement is alleged to have occurred. The Court allows subpoenas to be served in connection with such "John Doe" lawsuits, in which the lawsuit is filed prior to identifying the target of the lawsuit through IP address records. MIT sends you this notice because it is required under the Family Education Rights & Privacy Act (FERPA) to send reasonable notice when it receives a subpoena requested potential personal education records. The Court has been instructing MIT to give such notice and to wait 14 days after the notice is sent before producing the requested information. Contrary to sometimes popular belief, MIT can not just tell the other lawyers to shove it. MIT is required to comply with lawfully issued subpoenas.
All types come to you directly from MIT Information Services & Technology (IS&T), but both are in response to a communication that MIT has received from an outside party ('the man'). In this instance, MIT is not 'the man'. MIT does not monitor your computer or network usage looking for copyright violations. In most cases, 'the man' is one of several companies employed by a trade organization such as the RIAA (music), MPAA (movies) or ESA (software) to find and prosecute copyright violations. Detecting infringement is what these companies get paid for, so they work hard at it, and it is not safe to assume that they do not know about a particular file-sharing network or type of content. We also suspect that .edu domains receive the bulk of their attention due to the rich pickings to be found there, so you are in their prime hunting ground.
Although MIT is not trying to catch copyright violators, it is also not trying to protect them. Illegal filesharing is a violation of MIT's policies as well as the law, so MIT does act on cases that are brought to its attention, provided they contain enough information to appear genuine. It's worth noting that MIT's internal process is independent of the subpoena process. Just because you haven't received a Takedown Notice doesn't mean a subpoena that has been filed to discover your name isn't valid -- MIT may never have been informed of an infringement prior to receiving the subpoena.
If you get a Takedown Notice, the process is simple. Read the email carefully. It will contain the entire original infringement complaint. Disable any peer-to-peer file sharing software you might be using, such as Gnutella or KaZaA. If you got caught fair and square, remove the files. Even if you think the notice might be bogus, search your machine for the listed files and verify that they are illegal copyrighted works if they are present. If so, remove them before re-enabling file sharing. Then reply back to IS&T telling them that your machine is legal.
If you get a Subpoena Notification, it almost certainly means that a potentially expensive civil lawsuit is coming down the pipe. You should consult a lawyer as soon as possible, particularly if you believe that you may have grounds for quashing the subpoena within the 14-day window.
Don't just blow it off. If you don't respond to a Takedown Notice within 2 days, IS&T will cut off your machine's network connection. For a static IP address, this means shutting off your network drop. For a dynamically assigned (DHCP) address, this means blacklisting your computer's network adapter so it can't get an IP address on either the wired or wireless networks. Getting it reactivated requires more effort than just cleaning up the machine and responding.
Don't do anything rash. For a Takedown Notice, don't contact the complaining company. They don't know who you are and they don't need to know. For a Subpoena Notification, don't modify your computer's contents to try and remove the infringement. Many subpoenas come equipped with protection of evidence notices, so removal of the infringing files might expose you to felony criminal prosecution in addition to the civil suit you may already face.