Question: I boil rice often and then store them. Next day when I take them out of the fridge, they are lumped together in one structure. I can heat them and then press them to separate them.

The same even happens within the pan in which I have cooked the rice, if I leave it alone for some 10s of minutes once the rice is ready.

What do I do to keep the rice separate?

Answer: 
To get very separate rice I use a long grain rice, pre-rinsed and cooked at a 2 to 1 ratio of water to rice by volume.

There are three factors that affect the tendency for rice to stick together:

* Length of the grain. Long grain is the least likely to stick together. Medium and short grain are starchier and are more likely to stick together.
* Amount of hydration. If your water to rice ratio is too high, it will cause even long grain rice to cake together.
* Cold rinsing. Rinsing your uncooked rice in cold water helps to wash away the excess starch. My basmati rice recipe calls for a rinse followed by soaking the rice in cold water for 30 min to an hour to ensure very fluffy separate rice.
I use basmati (long and broken) as well as Round (Long as well as the round variety) for cooking.
For basmati, I recommend following the soaking and right proportions.
For brown rice:
With 3:1 ratio of Water to Rice, cook until two whistles in a pressure cooker. When the pressuring is complete, take off the cooker. 
Open it.
Add cold water. Stir it once.
Drop it in a rice strainer. 
Wait until the water is off the rice. Serve hot. 
It makes fluffy rice without sticking at all. 
You can ferment it at the end of the night and eat it in the morning as Kanji (south Indian- Keralite brown rice breakfast delicacy with fish/ green gram dish + pappad). It's YUM!