Soutenu means supported or sustained. Most often, teachers use this term to describe a soutenu turn (assemblé soutenu en tournant), but the term itself can apply to a variety of movements because it describes a quality of movement rather than a specific step.
Assemblé soutenu en tournant, which many teachers shorten to soutenu, is a turn in which the dancer sustains the height of the demi-pointe or pointe while turning. The dancer draws the extended leg in to relevé in 5th. Sometimes the movement begins with the dancer performing a relevé on the supporting foot (especially at the barre) and other times, the dancer piqués or developpés onto demi-pointe/pointe and then draws the second leg in to meet the supporting leg. Once the demi-pointe/pointe level has been created, the dancer crosses to fifth and rotates the body such that the opposite foot finishes in the front. The quality of the movement requires that the dancer's legs come together quickly and that the dancer finishes the turn in demi-pointe/pointe before lowering to demi-plié.
In more general terms, soutenu can be used as an adjective to describe any step or movement that is held by the dancer before progressing to the next step. Most often, one leg is brought to meet the other and the two legs act together to support the body.
Soutenu is pronounced [sut(ə)ny]. Sout sounds like "soup" with the final "p" switched to a "t". The "e" sound in the middle is optional depending on your pronunciation and sounds like "euh." And nu is similar to the first two letters of "nuclear" but the lips are very round and the lower jaw distends slightly. (This u sound is one that we don't have in English). Resist the American temptation to add a W sound to the end of nu.
Imagery that I have found useful in executing and teaching soutenu en tournant include the following: the legs come together to fifth on demi-pointe/pointe to form a support post (imagine the supports hold up the roof in a large warehouse), imagine squeezing the legs together at the end of the turn to sustain the position for a moment before the demi-plié, or visualize a small block or saucer of tea on top of the head to ensure that the turn stays on the same level throughout the rotation.