Qatar observes several public holidays including the weekend days of Friday and Saturday but you should be aware that some companies operate Saturday thru Thursday so it is a good idea to check your working hours before signing that contract.
Just so you understand a little bit more, Muslim festivals are timed according to local sightings of various phases of the moon so these dates are approximations. During the lunar month of Ramadan that precedes Eid al-Fitr, Muslims fast during the day and feast at night and you will notice that your working hours are shorted and business will open in the evening often staying in operation until midnight. Eid al-Fitr and Eid al-Adha are big vacation times for locals and expat as companies maybe closed for anywhere from 2 -10 days.
14 February 2013 - National Sports day
07 August 2013- Eid al-Fitr (End of Ramadan)
15 October 2013 - Eid al-Adha (Feast of the Sacrifice)
18 December 2013 National Day