A chemical reaction takes place when milk is mixed with vinegar. Milk contains protein, out of which about 80% is known as casein and the rest is called whey protein.
When milk is mixed with vinegar (an acid), the casein separates out and can be removed as solid lumps. The precipitate is nothing but denatured casein protein. The resulting solid lumps are bendy and pliable. This is because casein is a very long molecule and can bend like plastic. Interestingly, if we add a small amount of baking soda to these lumps, they will become sticky (like a glue).
One can also say that vinegar causes coagulation of the casein protein present in the milk resulting in the formation of the solid lumps, which then precipitate out of the liquid phase.
It is interesting to note that the reaction is faster if the milk is hot. The reaction between milk and vinegar is also known as curdling and can be beneficially used to produce cheese.
Hope this helps.