At home level, garbage is disposed by 3 ways -
a) Either it is collected from your door by municipality or a private contractor / sweeper
b) Home owners throw their garbage in a community garbage dump
c) Home owners dispose their garbage wherever they can.
The private contractors, who are supposed to take the collected garbage to the dumping grounds, usually don'd do it and dump it in open spaces (like this one -http://cj.ibnlive.in.com/slideshow7815.html). Worst part is that they aren't fined by authorities.
Municipal authorities mark certain grounds/wastelands away from populated areas as dumping grounds, dump the trash there and let it decay. (like this one - http://images.businessweek.com/ss/09/08/0805_biggest_garbage...)
Also, Indians, in general, while traveling or commuting or walking, dispose paper wrappers, plastic plates, flowers, water bottles, chewing gums or any other garbage on road or in public places.
For example, it's a common site to see a family having ice creams or chips or peanuts or soft drinks in parks. But when they leave, they leave the waste as it is. Even when there is a garbage can in sight. (see this - http://rhetoric2reality.files.wordpress.com/2010/11/public-p...)
An example of trash thrown by train travelers - http://3.bp.blogspot.com/-BA99LRmPyaA/TYsZstie7fI/AAAAAAAACC...
Heck, we can't even keep our places of worship clean. These are flowers, incense sticks and other offerings thrown in Ganga river after worshiping (http://3.bp.blogspot.com/-cO-PjFR2Kcw/TjgKYEIQiFI/AAAAAAAABG...). Ain't this ironical? Indians can pray and pollute a same object.
Coming to the decaying and non-decaying part - Most of it is decayable. However, the sheer volume of it does not allow nature to work. Hence it starts rotting and stinking. Also, stray animals (cows,dogs and pigs) dig through the trash mountain to find anything edible and in turn spread it more.
Having said that, things are improving. I have seen many young people (my friends included) carry the plastic trash or water bottles with them till they find a garbage can. Also, a few educated people are running cleaning campaigns in their cities (http://www.facebook.com/pages/Clean-Vadodara-Green-Vadodara-...). Also, shopping malls and large stores have started to charge for plastic carry bags. This has reduced plastic waste by bit.