A Kingdom is a sovereign state instituting a monarchy, or having a monarch as its head.
There were several kingdoms in Middle-earth, the largest being those of Rohan and Gondor. However at the time of the War of the Ring Gondor was governed by a Steward rather than a king, until Aragorn's return, and the re-establishment of the Reunited Kingdom.
There was also the Kingdom of Rhovanion
Kingdoms are dominions that are ruled by a King or Queen or both. There are generally monarchies. In the case of the Kingdom of Gondor in the Third Age it is not ruled by a King or Queen but by a steward but it is still a monarchy because the steward still rules just like a King or Queen only the steward is a replacement until the heir of Isildur claims the throne. A real world example of a Kingdom could be England in the 1700s or Spain in that same time.
Most Kingdoms have the desire normally given by the ruler to expand there territory. Clearly seen in the desire of Mordor and the Harad to expand in western territories belonging to the free peoples of Middle-earth.