The year people finally bought the theory that the Earth revolves around the Sun and not the other way round, which was suggested by Nicolaus Copernicus, sometime in the 16th Century, probably was when the seeds of this question were first sowed. The answer is very simple: the Earth moves around the Sun alright, but its axis is tilted.
Why Do We Have Different Seasons on Earth
A tilted axis? Now we all know that the Earth rotates around its own axis and at the same time, revolves around the Sun, but unlike what most people may have pictured, the axis around which the Earth spins around isn't exactly vertical. The axis on the north pole remains tilted at an angle which is estimated to be around 23.5 degrees more than a perpendicular, and stays that way through the revolution path. So, along the Earth's revolutionary path around the Sun, the northern hemisphere is closer to the Sun for sometime, and then when it completes half the revolution, the southern hemisphere comes closer to the Sun. So basically, the Earth's axis remains tilted in the same direction all through the revolution which causes the changes in seasons.
The catch here is that because of the tilted axis, the two hemispheres never have the same season at the same time. When one hemisphere has a particular season, the other one has a different season altogether. So, let us see the positions of the Earth and the different seasons experienced during the different months of a year.
December - February
In these months, due to the tilted axis of the Earth, the southern hemisphere is closer to the Sun, while the northern hemisphere is farther from the Sun. Hence understandably, the former will be warmer. So, it will have warm summer during this period, while the latter will have winter.
March - May
This is the first set of transition, where the northern hemisphere starts coming closer to the Sun, and the southern hemisphere starts drifting away from the Sun. So it is like the warming up period in the northern part, which gets successively warmer, causing spring in countries present there. The southern part, on the other hand, starts cooling down, causing autumn in countries present there.
June - August
In the months between June to August, the northern part starts coming closer to the Sun, with the axis at the north pole pointing towards it. The southern part on the other hand moves further away from the Sun, hence these months see the former having summer while the latter having winter.
September - November
This is the second set of transitional months, where the northern hemisphere starts moving further away from the Sun while the southern hemisphere starts coming closer to the Sun. The countries in the northern part start cooling down, while the countries in the southern part start warming up. Hence during these months, the northern countries experiences autumn, while the southern countries see spring.