A system whereby a set of rules are imposed and enforced upon a population. Government determines who holds the power, and who does not. This might be contrasted with a state of nature or anarchy. But this contrast would be hypothetical, as at this point in history government rules all.
A government may enforce morality, positive rights, negative rights, social ideals, or itself. One thing a government always looks out for is its own strength and power. This is sadly necessary, as there are many more people desiring political power than can actually hold this power, and many of these people are quite ruthless.
There are many types of government. Americans swear by democracy, a system where the citizens have input on the formation or implementation of laws. A citizen may be defined as any set of people, not bound to the set of people ruled by the government. While democracy is gaining popularity around the world, direct democracy is still virtually unheard of, abandoned in favor of representative democracy. Other flavors include participatory democracy, social democracy and Jeffersonian democracy.
If you're a realist, you might prefer an aristocracy, that is, rule by the best suited. Unfortunately, since power is always abused, aristocracies tend to quickly devolve into rule by the powerful. You end up with a closed group of people, the members of which are usually determined by genetics, wealth, and power. There is also aristodemocracy. A form of government where the power is divided between the 'great people' of the nation and the masses.
A surprisingly popular and successful strategy is Monarchy, where power is held by one person, the king (or queen). The ruler is usually decided genetically, with occasional wars weeding out the weak and the unfit.
Socialism is not a type of government, exactly, but Communism is socialism supported by the government (or, in all cases so far, just state capitalism). It sounds like a good idea, but darn it, we just can't get a country going without a good free market to support it. (Props to the USSR and China for making a good go of it, tho.)
Fascism is a one party government, usually with very strict control over the populous. Democracies and Communist regimes can be fascist without loosing their democratic/communist status. Strangely, while monarchy has generally positive connotations, fascism does not. This is probably due to the influence of Adolf Hitler and Benito Mussolini.
Bureaucracy is not really a form of government, but it deserves mention as the primary manifestation of modern governments.
See also:
Autocracy--Power is controlled by one person or a small group.
Bureaucracy--Many officials, rules are carefully followed.
Chrysocracy--Rule by the rich.
Demonocracy--Government by demons (not necessarily literaly).
Despotocracy--Rule by despots.
Diarchy, Dinarchy and Duarchy--Rule by two people.
Doulocracy or Dulocracy--A government by slaves.
Gerontocracy--Government by the old.
Gunocracy, Gyneocracy, Gynocracy or Gynecocracy--Government by a female or females.
Hagiarchy and Hagiocracy--Government by a priesthood.
Heptarchy--Rule by seven people.
Heterarchy--Government by a foreign ruler, or joint rule with a foreign state.
Hierocracy--Government by ecclesiastics.
Jesuitocracy--Government by Jesuits.
Kakistocracy--Government by the worst.
Kleptocracy--Government by thieves.
Kritarchy--The rule of the judges over Israel.
Meritocracy--Power is given to those with the most ability.
Mobocracy--Rule by the mob (masses).
Monocracy--Rule by one person.
Neocracy--Government by the new or inexperienced.
Nomocracy--Government in accordance with a set of laws.
Ochlocracy--Rule by the multitudes.
Oligarchy--power is in hands of a few persons.
Pantisocracy--All ruling equally.
Paparchy--Government by a pope.
Pedantocracy--Rule by pedants, or strict rule-bound scholars.
Pentarchy--Rule by five people.
Phylarchy--Government of a class or tribe.
Plantocracy--Government by the planters (farmers).
Plutocracy--Power in the hands of the rich.
Polycracy--Rule by many.
Pornocracy--Rule by harlots.
Ptochocracy--Rule of beggars or paupers.
Slavocracy--A term referring to all slave owners. Not a government.
Statocracy--Government by the state (as opposed to ecclesiastical power).
Stratocracy--A military government.
Synarchy--Joint rule.
Thalassocracy--Sovereignty of the seas.
Thearchy or Theocracy--Government directed by god.
Timocracy--Government based upon love of honor, or ruled by the land-owners.
Triarchy--Rule by three people.
Whiggarchy--Government by Whigs.