Why is air pressure higher in winter than in summer? At the top of a mountain, say Mt Everest, atmospheric pressure is low.
So shouldn't the same thing be true for winter season.
I.e air pressure in winters should be lesser than that in summers.
But it's the opposite.
Can someone please explain why ?
 A: 
At the top of a mountain, say Mt Everest, atmospheric pressure is low. So shouldn't the same thing be true for winter season.

Atmospheric pressure is low at the top of a mountain because the top of a mountain is at a high altitude. A nice first order approximation of the atmosphere is that the atmospheric pressure at some point in the atmosphere is the equal to the weight per unit area of all of the atmosphere above that point. This immediately leads to a roughly exponential decrease in pressure with increased altitude. Extending this first order reduction in pressure with increased altitude to a reduction in pressure with increased latitude is invalid logic.
On to the main question:

Why is air pressure higher in winter than in summer?

This is not the case. Suppose you pick a location in a temperate or polar climate (roughly between 30° and 70° degrees latitude) with a very long history of meteorological records. If you carefully compute the average atmospheric pressure in midwinter versus midsummer, you will likely find that the average pressure is very slightly lower in winter than it is in summer.
What you'll also find is that the extremes of pressure are more likely to occur in winter rather than summer. A nice example of this is wunderground.com's compilation of U.S. city barometric pressure records. All of the highest pressure records occur in winter. However, with the exception of coastal cities that were hit by hurricanes, all of the lowest pressure records also occur in winter.
This suggests that wintertime weather is much more volatile than is summertime weather, at least in the contiguous U.S. This concept applies across locales with temperate or polar climates: In such locales, wintertime weather tends to be much more volatile compared to summertime weather.
There's a marked temperature difference between the Earth's equatorial and polar regions. The Earth's atmosphere works to balance these differences via air flow from the equator to the poles and back.  The primary cause of the increased wintertime volatility in temperate and polar locales is that this is when the temperature difference between the equator and the pole is at its greatest and that these are the locales where the conflict is most brought to bear.
A: Air pressure is lower at the top of Mt. Everest because there is less atmosphere above it to compress the air, making the pressure less.
I can't be sure what you mean by "air pressure in winters should be lesser than that in summers", I have never heard of this being a straightforward observation before. But (to my knowledge) it is probably because where you live, winters tend to be drier. Dry air is denser than moist air (water molecules are lighter than nitrogen and oxygen molecules. When it is humid water molecules displace nitrogen and oxygen molecules making the air lighter). Also, cold air contracts, and is therefore more dense than hot air. 
A: One of the reasons the air pressure is higher in winter is because the air is colder so the molecules stick closer together building more pressure. Also Mt. Everest is the highest elevation above sea level so it would seem like it would be cold, BUT there are many things that factor into it that make that statement wrong. One of them is that warm air rises because it is less dense the hotter it gets. Hope this helps! This website has helped me a lot on this subject.
Here's the link.
http://www.wxdude.com/singalongcompanion/windandairpressure/index.html
A: The pressure profile of earth from the surface all the way to the last layer of atmosphere is a decreasing/negative gradient, why is that? It's gravity. The strength of gravity on a mass is inversely proportional to the square of distance away from the gravity source. Air layers closer to the planet weighs more than layers farther because the planet is pulling those air molecules stronger than those farther above and also all the rest of the atmosphere from that point up is weighing down on anything at that point. In terms of temperature, temperature isn't much of a factor that affects gravitational pull, but since gases compress/contract when colder, we can say there's more air weighing down per $m^2$ than before. This last point I'm not so sure.
