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: 
So I know of two ways, one way is what happens with commercial Cantonese cooking:

1. Add rice and water to your rice cooker, and throw in a knob of butter, diced, mix it all amongst the rice so it will be vaguely even;
2. Cook rice as per normal.

Reasons: adding fats help to keep rice from sticking. They do this so they can make fried rice with fresh rice, rather than waiting for rice to stale for 1+ days in the fridge.

The other way is from Persian cooking:

1. Add rice and water to a pot and par-boil;
2. Once par-boiled, remove and steam the rest of the way.
	1. Sometimes they will add some oil too.

Reasons: possibly adding fats too, but also, not oversaturating the rice with water will mean they are harder, and can retain their shape better.

Just a bit of a different take to @Chee's Burgers' answer.
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!