Fixing a slowly leaking toilet tank
The part that you most likely need to replace is known as the flapper. It is the part shown in your picture that has the hinge arms that go back to the posts on the sides of the stand pipe. The flush lever also connects to this flapper to pull it up when the toilet is flushed. 

The flappers are made of a flexible rubber or silicone material. Water treatments like chlorine in the water will degrade the flexibility of the flapper over time. This can lead to cracking or hardening of the flapper and can allow for the type of slow leak like you describe.

I find that I have to replace the ones in my house about once each 2 years or so. Note that there are some types that are better than others so look carefully at the options that you find at the home center or hardware store. Some of the replacement parts are designed to be "universal" and can be adapted to the several different hinge styles used in the water closet of the toilet. The ones I purchase generally have to have part of the thing cut away to adapt it to my usage.

Take your picture with you to the store. It will help you pick out just what you need.