Dance partners stay together for the duration of the dance and dance independently of other couples dancing at the same time, if any.
Although this kind of dancing can be seen, e.g., in ballet, this term is usually applied to various forms of Social dance and related forms: Ballroom dance, Folk dance, etc.
Partner dance may be a basis of a formation dance, a round dance, or a sequence dance. These are kinds of group dance where the dancers form couples and dance either the same routines (pre-choreographed or called) or routines within a common choreohraphy that controls both how the each couple dances together and how each couple moves in accord with other couples.
In many partner dances, one, typically a man, is the leader; the other, typically a woman, is the follower. As a rule, they maintain connection with each other. In some dances the connection is loose and called dance handhold. In other dances the connection involves body contact. In the latter case the connection imposes significant restrictions on relative body positions during the dance and hence it is often called dance frame. It is also said that each partner has his own dance frame. Although the handhold connection poses almost no restriction on body positions, it is quite helpful that the partners are aware of their dance frames, since this is instrumental in leading and following.
In promenade-style partner dancing there is no leader or follower, and the couple dance side-by-side maintaining a connection with each other through a promenade handhold. The man dances traditionally to the left of the woman.
Some peoples have folk partner dances, where partners do not have any body contact at all, but there is still a kind of "call-response" interaction.
This kind of dance involves dancing of three persons together: one man with two women or one woman with two men. In social dancing, double partnering is of choice when a significant demographic disproportion happens between the two sexes. For example, this happens during wars: in army there is lack of women, while among civilians able dancers are mostly women, especially during enormous wars such as WWII.
Today (1980-2003), double partner dance is often performed in Hustle, Salsa and Swing dance communities, experienced leaders leading two followers.
There are a number of folk dances that feature this setup. Among these are the Russian Troika and the Polish Trojak (Wikipedia link in Polish) folk dances, where a man dances with two women. A Cajun dance with the name Troika is also known.
See also:
Double partner dance