Keep a regular check and gather information on some matter which is very important to you.
Keep a regular watch on somebody who is very important to you, whom you may suspect for something.
1. Shyam, please keep an eye on the stock of "New Offers" or "Cash discounts" available in your store and let me know immediately.
( Here, the information on New Offers and Cash discounts are to be notified)
2. I asked my husband to keep an eye on our kids when I am away on tour.
(Here, my husband was asked to watch the activities of our kids to see that they are disciplined properly)
3. I found cash missing from my drawer; now I have to keep an eye on my maid.
(here upon suspicion, maid needs to be watched.