Why is oxygen so special?
Dioxygen, $\ce{O2}$ is a very special molecule. A good majority of organisms on earth use dioxygen to breathe and survive. Oxygen is also the second most abundant element in the sky, as well as the most abundant elements in the earth's crust. What makes it so special? Its because of its abundance, which is obvious, and also because of it's strong oxidizing power. 

If you are a student below 11th grade, you might be taught that oxidation is addition of oxygen or the removal of hydrogen. I will now, introduce a new definition of oxidation. 

Oxidation: The loss of electrons from a substance is called oxidation.

The reagents that cause substances to become oxidized are called oxidizing agents. $\ce{O2}$ is an example of an oxidizing agent. A popular oxidizing agent in analytical chemistry is Potassium Permanganate ($\ce{KMnO4}$ ).

Many times, an actual loss of electron doesn't take place, rather a shift of electrons from one element to another within the same molecule occurs, which is also considered as oxidation. To explain such stuff, we introduce the concept of oxidation numbers. An easy way to find oxidation number of an element in a compound is:

Ox.num=(Num. of bonds formed with electronegative atoms)-(Num. of bonds with electropositive atoms) 

For example, take carbon dioxide. It's structure is:



Note that bonds made by an atom with a similar atom are not considered (like $\ce{O-O}$ or $\ce{S-S}$).

Carbon makes four bonds with more electronegative atoms (oxygen). So it's oxidation number is +4.

If oxidation number of any atom of a molecule is increased through a reaction, we say that oxidation has taken place. Note that as this happens, the oxidation number of the oxidizing agent will decrease. 

Now, finally (with a deep breath), lets come to your question. 

Speciality of Oxygen:

Oxygen in nature, as dioxygen has a 0 oxidation number. Since it's so electronegative, it almost always forms bonds with electropositive atoms, and so its oxidation number in compounds is always negative. Oxygen itself has such a high tendency to rip off electrons from other molecules. This is what makes oxygen very special. 

There are so many examples of oxidation reactions with oxygen:

Combustion: Burning stuff itself is an oxidation reaction. If you burn coal(carbon), its oxidation number  goes from 0 (in pure carbon) to +4 (in carbon dioxide). 

Respiration: Even our own bodies have to do an oxidation reaction to keep us with energy to do our daily work, like asking and answering questions on StackExchange. Glucose, which has a oxidation number of 0 (equal number of bonds with oxygen and carbon in its formula of $\ce{C6H12O6}$ ) is oxidized by the oxygen we breathe to carbon dioxide. 

Much of the interesting properties of oxygen is due  to its oxidizing power. While many oxidizers are known, like fluorine for example; oxygen is, and will always, be the most well known one!