Military ?

Deployments can be anywhere from a week to a year. Most of the time it is a whole unit/command that departs their home base/post and goes somewhere else for a period of time before returning back home. You can be in one location or move to multiple while deployed. In the Navy a WestPac/MedRun are deployments. While deployed you will continue to get BAH based on where you are stationed but you can be eligible for other pays and allowances depending on where you are deployed to. 

TDY is Temporary Duty, you usually by yourself to a location and report to a new command/unit for up to 189 days before returning back to home. You will get per diem to cover housing and food while you are there. This will be as well as the BAH you are getting for home. 

Length.  Typically over six months is a deployment.  Under  is TDY