The valve that releases the water from the tank into the bowl is letting water slip by, which is going right down the bowl and out of the toilet. As the water level slowly goes down in the tank, at a certain point the float mechanism will activate the siphon to refill the tank. This is the noise you are hearing.
Reasons why your toilet may not be flushing:
If you have an old fasioned lever flush toilet, it is most likely to have a Syphon for flushing. The Syphon contains a pvc sheet called a Diaphgram and this is required to create a vacum and lift the water required for flushing, if this is split or perished, it cannot function.
The flush handle may have worked loose from many uses and is not turning the link chain sufficiently to lift the diaghram.
The Link between the handle and the syphon is twisted / stretched / broken and is failing to lift the syphon enough to create the vacuum needed.
How to reslove these problems:
The simple, and cost effective answer is to replace the old, outdated syphon with a fleximount Flush Valve,
Not only will this be far more reliable for flushing, but it will give you the following benefits.
Convert an old single flush toilet to a Dual Flush.
Save water, and if you are metered, money.
Convert an old toilet with an external overflow into an internal overflow.