Government can be defined in a myriad of ways. The most elemental definition is that it is the authority that rules over a country's body politic and serves as ultimate arbitrator in disputes between individuals. Beyond this, we get into some very contested aims of government, its ability to intervene in matters, its focus, and what it hopes to do. Philosophy is critical in understanding forms of government. The different forms of government reflect the philosophy that advocates it. To define all aspects of government would take longer than the space provided here. Examining some of the sites below might help you in being able to identify different forms of government. For example, someone who believes that government should not be present would follow an "anarchy" style of government where there is government by none. In contrast to this, a "communist" form of government is where the state/ political body controls all aspects of the social, political, and economic order. A "democracy" is a type of government where the people have ultimate power over the government, and a representative democracy is a form of government where this power is voiced through elected representatives.
Can you imagine for a moment what your life would be like if there were no government? That would mean there was no one in charge of your city or town, no one in charge of your county, no one in charge of your state, and no one in charge of your country, since all of the institutions in charge of those areas are forms of government.
You would not have to pay any taxes to anyone, which might seem great for awhile, but you would have to be responsible for building and repairing your own little section of the street, you would have to pay for schooling for your children, you would have to contract with someone to have your rubbish picked up, you would have to figure out a way to get your own safe water supply, you would have to pay for your own police and fire protection, and you would have no one to complain to if your neighbors played their radio too loudly or trampled your lawn. You would not be able to sue anyone who did some wrong to you, and you would be driving on roads that had no speed limits or traffic rules, even if you had any roads at all, since government is responsible for building roads. It might seem like fun to be able to go as fast as you wanted and not have any red lights to worry about, but after awhile, you would figure out this was not the best situation. You would have to solve many problems on your own. For example, if someone stole something from your house, it would be up to you to figure out what to do about that. If your dog, which would not need to be registered or have up-to-date shots, bit someone, that person would not be able to call on government for a solution, and might just come over to shoot your dog.
I could go on forever with the list of functions government is responsible for, but perhaps you can think of some other functions for yourself and decide what all of these functions have in common.