For a 1-person city far from other cities, it covers a 4-tile radius (3 tiles diagonally). As the city grows, the territory expands. Its border has a nation-specific color, noted on the nations menu (F3). For a single human playing against AI nations, the human's border is red in the default ruleset.
In-game help related to territory can be found under "National Borders", a subheading under "Terrain". Part of that says "Once claimed, a tile that can be directly worked by a city can not change ownership unless the city does (or is destroyed)", but that is wrong in 2.5.0: if a foreign city grows bigger or is built very close, its territory may expand to steal good tiles that the slower city was working.
The territory, at least up to 2.3.5, did not normally include water except immediately adjacent to a city. However, territories in 2.5.0 can extend to cover some nearby water even though the in-game "Help" still says "Borders can only extend into water for tiles adjacent to a city; other water tiles remain unclaimed territory".
Options at the start of a game include "no borders" and "see everything inside border". You can also ask for border changes to show even under fog of war, which indicates when a city you once saw has changed hands and which may allow you to deduce that a city has been destroyed.
End of armistice
In the turn on which an armistice changes to a peace, units that should not be in a territory are disbanded.
End of alliance
If an alliance is cancelled, foreign units in an allied city are immediately disbanded but elsewhere in the former ally's territory they may move out.
Effects of enlargement
If your territory enlarges because of city growth, the enlargement is generally no more than one tile-width per turn. That usually allows units with which you are at peace to move out if they are then in the enlarged territory. However, a major enlargement caused by a new city-build is often impossible to escape.
If military units with which you are at peace cannot move out of the enlarged territory, they are virtually frozen. Any attempted move to another tile in your territory, except to share a tile with a fellow-military unit, is prohibited as an attack. The only solutions are a change of state to war, an alliance, or a form of government that can ignore territorial restrictions under peace.