Question: Few months back I purchased a second hand tripod.  

Later on I noticed that the head *(which holds the camera)* of the tripod doesn't move freely.  

I mean I have to apply some pressure to make it move left and right, and even when it moves left/right it jumps some fixed *x* degrees, i.e I can't make it move *x+1* degree, it moves straightaway to *x+5* degrees.

I don't know whether this is normal or not, but this is very annoying.

Is there a way that ***I*** can fix it? Perhaps by oiling at some right spots?

Answer: 
Quintessentially I think that the problem is that you've got a cheap tripod. In order to have a smooth pan and tilt you need to get an expensive tripod head suitable for film and video work.

Because the cheaper heads work on friction you can make them move smoother using oils or other lubricants, however this will also prevent them from locking up completely meaning that you may get some drift at some point or even, if unlucky, render the head unusable until cleaned.

There isn't really a way around the problem as far as I know. Cheap heads are stiff and jerky while expensive heads aren't.
Is this when the controls (knobs/handles) are quite loose? Is it a ball head, or a pan/tilt type head?

Your best bet would be to try to disassemble it, if possible, and clean it. If you can't get it apart, but can access some surfaces, clean them with a microfibre cloth.

You can put a small amount of oil or silicon lubricant on it, but that may cause further problems

* may attract more dust/dirt, which would adhere to the oil
* may make it harder to lock down in place, so the head slips after you tighten it

If this is a large tripod intended for a large SLR and you have a small point and shoot, it may be that the lack of weight makes it hard to make micro adjustments like you want. If the head can be replaced, keep the legs and buy a small ball head for your P&S.