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 simply put the rice in a jar of water and shake like heck to rinse well i then use the two to one method and add a tiny bit of salt but most importantly a tablespoon of olive oil and another important thing to do is stir often . Another way i have heard of that i thought was crazy until i seen it done a couple of times is to put a wet teatowel on top of the pot. I am to scared of doing it this way as i dont want to burn the teatowel i am not sure but i think this is a middle eastern method ? But my freind cooked it like this once when he made a dish for me and him once and i have never had better rice