Question

What kind of orbit in which the satellite stays in the same location in the sky

Answer 1

That's called a "geostationary" orbit.  (Not "geosynchronous".)

It's the situation when the satellite's orbit is

-- perfectly circular,
-- exactly over the equator,
-- and at exactly the right distance from the center of the Earth
(42,164 km  or  26,199 mi).

None of these conditions can ever be perfect, and in fact they change over time.
So every real geostationary satellite actually describes a little figure-8 in the sky
once a day.  The better its orbit is, the smaller the figure-8. 

In order to keep the orbit reasonably correct, it needs a puff from its onboard
thrusters every once in a while. 

That's how a TV or weather satellite reaches its "end of life", if it doesn't get
smashed by a meteoroid first:  Either its solar panels get too dusty and chipped
by micrometeoroids to generate enough power, or else it runs so low on fuel for
its onboard thrusters that it can't be kept in a useful orbit.  At that point, the fuel
that remains is used to intentionally put it into a crazy orbit, where it will dip low
into the atmosphere and burn up. 

Why destroy it ?  To get it out of the geostationary orbital track, and open up one
of those precious slots for a new, young satellite to take its place.  There are only
a few hundred slots available where a satellite can be geostationary.  (And a slot
over the Indian Ocean doesn't do TV viewers in the US much good.)