I'm replacing the kitchen sink and noticed this part just past the p trap on one of the 2 drains (the side with the disposal). Any idea what this is and what is its propose?
It is called a drain saddle clamp. It is added when there is no accommodations to have an added appliance like a Reverse Osmosis system as suggested by Mike, needs a way to remove water as part of its function.
The one you have looks like it has been capped, as if the RO system has been removed. If that is the case, you can replace the white plastic part of the drain that has it.