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: 
Force some talcum powder (or better yet, powdered graphite) into the friction surfaces of the head. This works like little ball bearings for the plastic-on-plastic rubbing of a cheap tripod head. Don't use oil, it just makes it worse.

I have a $25 Amazon Basics tripod, and this made a big improvement. I drilled a little hole in the panning head to force the talcum powder in, but maybe I could have just rubbed it into the cracks. It made a big improvement in the stiction and judder. The only downside is that it now smells like baby powder.
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.