In Civilization IV, a Vassal State is a civilization that is, in a sense, "owned" by another civilization, though they still have control over their territory. The human player's civilization most likely cannot become a vassal, as the option to do so is never available.
Much of the information here comes from a CivFanatics info page by Crighton.
A civilization can become a vassal of another if the first civilization is at war with the second and capitulates, if the first civilization asks to become the second's vassal for protection, or if the second civilization (most likely controlled by a human player) successfully convinces the first to become a vassal. If the master or "owner" civilization declares war on another, the vassal automatically does so as well. Colonies are similar to vassal states, except colonies do not exist until their Mother Country (like masters in a regular Vassal State agreement) grants them independence. Normal Vassal States exist before becoming vassals.
The master enjoys complete freedom of movement in the vassal's territory, including the ability to heal normally in the vassal's territory and use fortifications owned by the vassal. The master can also investigate any vassal city, and as of the v2.08 patch the master can now airlift units directly into a vassal's cities. The master no longer has to pay supply costs for units inside of the vassal's lands. However, the master must agree to an Open Borders agreement with the vassal in order for the vassal to move units into and out of the master's territory.
The master can demand any SINGLE resource from a vassal, even one that the vassal is using. The vassal has the right to refuse the SECOND demand, but this will immediately plunge the two civilizations into war. This can come in handy if a vassal has the only strategic resource available to its master (Copper, Iron, Horses, Oil, Aluminum, etc.).
The master's people enjoy increased happiness. Those in the vassal empire suffer decreased happiness, which acts as an incentive for the vassal to purchase luxury goods from the master to placate the populace.
The vassal can't make war or peace on its own. It immediately adopts the master's war and peace relationships.
Half of the vassal's territory and population count towards the master's domination victory AND score, which can truly speed up a Domination or Conquest win.
The vassal will always vote for the master in world leader elections (unless the vassal itself is a candidate).
Over time, the master can rehabilitate his or her relationship with the vassal through normal diplomatic means (Open Borders, Shared State Religions, Mutual Military Struggle, occasional gifts, etc.). Also, after the v2.08 patch the Mutual Defense Pact bonus is now added into the diplomatic modifier between the master and the vassal.
Having vassal cities will incur higher maintenance cost for the master's own cities.
Other civs may like the master of a Vassal State a little less. The master will most likely see a -1 Diplomacy Penalty in his or her relations with other civs, though this penalty may change depending on the number and power of vassals.
Similarly, a vassal who has gone around the continent stirring up trouble is not going to be a good choice to become a Voluntary Vassal, as the other civs will view the master's patronage as an alliance. The AI hatred of a vassal may override their friendly feelings towards the master.
Although the master can direct what a vassal researches, the master doesn't share in the discovery. If a vassal somehow manages to acquire some technology the master doesn't have, it doesn't have to and can't be forced to share the technology. This is the complete opposite of the same occurrence during a Permanent Alliance, where the two allies can research different techs and the discoveries are shared between the two.
A voluntary Vassal State can declare autonomy at any time, though its master may choose to go to war with it in response. A capitulated Vassal State can regain its autonomy under one of three conditions:
The Vassal State loses more than half of its territory that it owned at the time it became a Vassal State. Note that any cities or territory a vassal loses to its master through culture flipping also count towards the loss of territory condition.
The Vassal State grows in size AND population to a point where it has more than 50% of the master's size AND population. This is a two-part condition, so both parts have to be met in order for the vassal to regain its autonomy in this manner.
The master demands a particular resource and the Vassal State refuses the master's request. This ends the vassal agreement and puts the two civilizations at war with each other. Since the vassal cannot refuse the master's first demand for a resource, this situation is not likely to occur unless the master is reckless or heavy-handed with resource demands.