Tactical Sprinting in Modern Warfare is a method to move faster, and while using this spring your weapon will be pointed upwards. This is the COD equivalent of running with a knife, and just like in real life, you are able to move slightly faster by not pointing the gun in front of you.
It is very easy to use Tactical Sprinting in Call of Duty: Modern Warfare. If you are a console player, then all you need to do is to double tap the thumbstick when you are about to run. This puts you into Tactical Sprint mode until you stop running.
If you play on PC, the default key for going into Tactical Sprint mode is the Shift key. Just tap this while running, and you will be in Tactical Sprint mode until you stop running, or run out of breath.
Tactical Sprint is not a permanent thing, and you can only do it for a little while before you will be tired and have to go back to normal running.
You should be using Tactical Sprinting when quickly dashing between two covers or every once in a while while transversing longer distances.