Start with the flapper
The flapper is the rubber seal at the bottom of the tank. It lifts when you flush and drops back to seal the tank. Rubber hardens and warps with age and with the chlorine in municipal water, and once it stops seating the tank leaks continuously into the bowl. The refill valve then runs to top the tank back up, which is the sound you're hearing.
To test it: put a few drops of food coloring in the tank and leave it thirty minutes without flushing. If color appears in the bowl, the flapper is the problem.
Check the fill valve and float
If the flapper seals but the toilet still runs, the fill valve is likely overfilling the tank until water spills into the overflow tube. Look at the water level. It should sit roughly an inch below the top of the overflow tube. If it's at or above the tube, the float needs adjusting or the fill valve has failed.
Check the chain and the flush handle
A chain that is too short holds the flapper slightly open. A chain that is too long gets trapped under the flapper as it drops. Both produce a toilet that runs intermittently rather than constantly, which is why the fault seems to come and go.
When it isn't the tank at all
If everything in the tank is new and it still runs, the problem may be a hairline crack in the flush valve seat or in the overflow tube itself. At that point replacing the internals stops being economical and the toilet itself needs looking at.
What it costs to fix
A flapper and fill valve rebuild is one of the cheapest plumbing repairs there is, and it's a flat-price job. Compare that against the water it's wasting every month and it usually pays for itself quickly.
If the toilet has been running for weeks and you'd rather it just stopped, we handle toilet repair across Michigan with parts already on the truck. Call (888) 809-6756 any hour, or read more about Toilet Repair in Michigan.