What is the duty of leg umpire in cricket?

If you're refering to the umpire who stands to the side and in line with the batter's crease, he is there for run out appeals at that end.
More and more these days, however, the 3rd umpire (Who isn't on the pitch, and is the person who uses technology to review decisions for the on pitch umpires.) is used for this decision for more reliable results.

--I------I-- <-- This is the pitch

^this is the crease.
* Here is the 'leg umpire'.