Platform screen doors are doors installed on the edge of a station platform, in addition to the doors of any train which cares to stop there itself.

Advantages

Platform screen doors (PSDs) are usually made of glass or metal, and are usually sliding doors (pocket doors could also be possible, although I can't recollect seeing any). They have several uses;

• Less suicide.
  The doors prevent anyone from stepping off the platform without boarding a train at the same time. Unless there's a really big gap between the two, of course.
  
  
• Less heating.
  They prevent the loss of heat, resulting in less energy required to power the station heaters, resulting in cheaper electricity bills.
  
  
• They're cool, too.
  You can stare at these things and marvel at the wonder of synchronized technology.

PSDs are synchronized with the train doors itself. The train driver (or, even more technologically advanced, driverless trains.¹) opens and closes the train doors, and, via the wonders of technology, causes the platform screen doors to automatically open in synchronization, although with some margin of delay.

Disadvantages

As of writing, PSDs still have some disadvantages:

• Cost.
  The big one. PSDs cost a hell of a lot to install.
  
  
• Compatibility.
  All of the trains have to be compatible with the PSD system, or passengers aren't going to be able to board. (A member of staff on the platform can open them manually via a control board, but that kind of defeats the point.)
  Also, trains with 'pop-out' doors require a bigger margin between train and PSD, increasing the risk of the following situation;
  
  
• Safety.
  PSDs add to safety in many ways, but they have some concerns of their own. You might consider being approached rapidly by the edge of the train door to be a bad thing, but compare it to having the train doors and the PSDs close on you simultaneously while you're in between both.

Known Installations

• Many of the platforms on the London Underground Jubilee Line Extension.
  
  
• They're also featured in the Houston International Airport. Driverless, they connect the three terminals and the long-term car park.²
  
  
• The north-south metro connection at Gatwick Airport. Not really a 'train', but an example of the technology nonetheless. These were actually fully driverless trains, and more of a monorail. On my return flight, the thing broke down and a person had to walk down the platform (which actually ran all the way along the track), hop in the cab and drive it manually. I guess no technology is perfect.