Answer:
First you need to understand what milk is, other than a just a white liquid you get from mammary glands. Milk is an emulsion of fat, water, lactose (a type of sugar) and a mixture of proteins (loads of other stuff, but these are what we will talk about).The small fat globules are surrounded by a skin of phospholipids and proteins, which are chemicals that help them stay as small globules in the mix, rather than cumping together into a big pat of butter. If you beat the milk, that is exactly what happens - and is how you make butter from milk.
The water in the milk contains soluble proteins, which wander around by themselves in the liquid, and proteins which have one end which likes water, and one end which doesn't. Think of a match - a red head on one end, with a tail of wood trailing behind. In order for the "water hating" ends of the protein (the wood of the match) to stay away from the water they are floating in, these proteins arange themselves into globes called micelles. Kind of like a circle of cows in a field standing in a protective circle, with all of the tails in the centre and a circle of heads looking outwards.
The proteins that do this are called caseins, there are four different types and they make up about 80% of the weight of the total protein in the milk. The outer layer of these micelles is make up of a type of casein known as kappa-casein, and the kappa casein reaches out a bit into the surrounding liquid. Under an electrom microscope, each ball looks a bit like a little polystyrene ball (like you get in bean bags and the like), with a bit of a lumpy surface. The kappa-casein has a negative charge, and as like charges repel, all of the micelles stay away from each other - which keeps them in solution.
An acid is any chemical which loves to give (positively charged) proteins away, and the stronger the acid, the stronger the tendency to force proteins onto other chemicals (which usually breaks them down into simpler structures - this is the iconic "fizzing" you see in Hollywood movies involving acids.)
As you add acid to milk, say by pouring in lemon juice, or by letting bacteria turn the milk sugar lactose into lactic acid, more and more positively charged proteins are given to the negatively charged kappa-casein, the kappa-casein loses its charge and so the casein micelles begin to clump together. Eventually the clumps become big enough to see - which are the lumps we call curds, and the process is called curdling. Good if you want to collect the curds to make cheese. Bad if you want a smooth drink or sauce!
If you heat the milk up to at least 85 degrees C before you add the acid, then the kappa-casein reacts with the soluble protein (called a whey protein, becasue it doesn't mind floating around by itself in the watery whey) b-lacto- globulin. The result is a complex which makes the casein micelle surface markedly coarser, so if you now add acid (usually by allowing a "good tasting" bacteria to turn the lactose into lactic acid) the casein micelles clump into an open spongy gell. This sponge soaks up the liquid, and you end up with Yoghurt. Yoghurt is much more stable in cooking and acepting acid for this reason.
See some microscope pictures here: http://www.magma.ca/~pavel/science/Yogurt.htm