It sounds like a leaking flapper. Your toilet tank has a device that lets water in (the "fill valve") and a device that lets water out (the "flush valve"). Depending upon what type of toilet it is, it will have one of several types of flush valve. The most common is a "flapper" that is lifted when you pull the flush handle ("trip lever"), thus letting water flow from the tank to the bowl. The flapper floats in the water for a moment, and then closes to stop the water from leaving the tank and allowing the tank to refill with water from the fill valve. The flapper (or whatever device is blocking the flush hole) is rubber or similar, and will wear out after a while.
When the flapper wears out, a little bit of water will drip from the tank into the bowl. After it drips from a full tank for a while, the toilet will start to run just a smidge so that it can replace the water which leaves the tank, and then turn off. When it isn't running, a little water is still leaving the tank past the flapper, going into the bowl very slowly, and then dripping out of the bowl and into the drain pipe for the toilet. That is likely what you are hearing: the water dripping from the toilet bowl into the drain pipe for the toilet. (To put more color on it, the toilet bowl, when properly refilled, settles to a specific level. Any more water that dribbles in will also dribble out over the "weir", down the trapway, out the outlet, and into the drain pipe under the toilet, then into the vertical drain pipe in the wall, down to the underground main drain, and out to the sewer. While the toilet water level is settling and if there is a little leak from tank to bowl, then that little leak will dribble over the weir and down the pipe in your wall...drip, drip, drip.)
Do this test. Turn off the water to the toilet and see if the tank doesn't go down some while the water is off. (What we do is mark where the water level is in the tank with a pencil, then come back in a couple of hours and see whether the water level has fallen. Or leave it overnight and look in the morning.) If the water has drained down, most likely you need to replace the flapper or other device. Look in the tank and tell us what model numbers you see in there (or better yet post a picture of the inside of the tank), and we can help you determine what kind of replacement flapper (or other device) you need. Flappers cost less than $10, usually, you can get them locally, and they will take you less than ten minutes to change out with no need for tools. But there are all types of flappers for all types of toilets, so we need to know what you have so we can point you in the right direction.
Good luck and let us know.