Each Shabbat every synagogue gives aliyot for Torah readings starting with Cohain etc. What is the reason behind the aliyot and why are they done in the particular order. e.g. Cohain, Levi etc ending with Maftir.

The seven Aliyot represent the officers who would serve a king. Just as a human king would have seven officers standing at attention, so too we show our honor for the Torah by having seven people called to read from it.

In order to minimize disputes over who would be called for which portion, the Sages ordained that the first two Aliyot should be reserved for Kohanim and Leviim respectively.
The concept of Maftir and Haftorah is based on the following. There was a time when oppressors of the Jews prohibited them from reading the Torah in synagogue. As long as this decree was in effect, the people began to read from Neviim (prophets) in the synagogue instead. When the decree was rescinded, they continued to read from the prophets when they finished reading the Torah portion of the week. It was felt that it appeared disrespectful for someone to read only from the prophets and not from the Torah, so the one who was going to read the prophets would first read a few verses from the Torah as well.