This information was updated on March 2, 2014
New dates for visiting the Vasari Corridor have been announced - it is now scheduled to be open from February 7th to April 30th, 2014, but of course if you visit the official page tickets are not available!
There are currently two ways to see the Corridor - do it yourself as described here, most likely with an Italian speaking guide (except on Fridays as mentioned in the comments below), or booking with a tour company. The first option is cheaper - but will take some diligence in calling Italy, the second is more expensive, but much easier.
The official website of the "Corridoio Vasariano" is here: http://www.uffizi.firenze.it/en/musei/index.php?m=vasariano.
If you want to try to make a reservation, call immediately - the Italian country code is 39, and the number to call is 055 294883. The best part - tickets are only €10.50 plus a €4 reservation fee.
Since we are Friends of the Uffizi (I will post about this soon) our entrance is actually free, and we will only have to pay the 4 euro reservation fee on the day we visit. Good luck!
Alternatively, if the tour sells out, or you can not get through, there is a tour company (one of many actually) selling guided tours in English. This is a more expensive option, but it may be your last and best chance to see the corridor for years if it closes for renovations (as planned - but not done yet for a lack of funding).