Psionic Laboratories, or Psi Labs are a critical weapon in the fight against the aliens. Soldiers using this facility learn how to use their "mind-force" in combat. To be able to build a Psi Lab you will first have to capture a psionic capable alien and interrogate it. Once you have done that you will be able to research, and then build these facilities.
Each lab trains ten soldiers at once and at the end of the month soldiers without any Psi stats will now show them, and they will have 16-24 Psi Skill points. Soldiers who already had some Psi training will continue to increase their Skill. Just as importantly when you first train any recruit you will reveal their Psionic Strength which is a statistic you cannot improve with training, above 80 is good, above 95 is excellent. Note that your Psi Labs will continue to train your soldiers, even once they reach the combat cap of 100. For more specifics on Psi Lab training (and to avoid repetition), see Psi Skill - Improvement.
Try not to transfer soldiers if they are using the Psi Lab. Training for that month will be lost, even if the destination base has Psi Lab space available. You can use soldiers in combat whilst they are training, but if they die, you'll have an empty slot in your labs, which can't be filled until the following month.
Soldiers able to use Psi-Amps due to training in the Psionic Laboratory are a powerful asset, but this is countered somewhat due to the restrictive method by which this training is obtained.
Note that you cannot dismantle any Psi Labs at a base if even just one soldier is in training. So place them carefully, planning for future growth such as hangars, etc. Or edit your base with something like XcomUtil.
Construction Time: 24 days
Construction Cost: $750,000
Maintenance Cost: $16,000/month