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.
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.