taru-adjective, adverb taking the 'to' particle
Describes a serene, unhurried state of mind or manner. Often used in literary or formal contexts. Cross-referenced with 悠々, which shares the same meaning.
See also: 悠々
彼は優々とパイプをくわえていた。
He was leisurely holding a pipe in his mouth.
優々たる態度で質問に答えた。
He answered the questions with a composed attitude.
taru-adjective, adverb taking the 'to' particle
Describes elegance and refinement in appearance or movement. Less common than the calm/leisurely sense; often found in classical or literary descriptions.
彼女の優々とした立ち居振る舞いは、まるで絵画のようだった。
Her graceful bearing was just like a painting.
悠々 is a more common spelling for the same word, especially for the 'calm/leisurely' sense. 優々 is a variant kanji form that can also carry the 'graceful/refined' nuance.
The exact derivation is uncertain. The kanji 優 (gentle, superior) is used here for its associations with elegance and composure, while 悠 (distant, leisurely) is the more common spelling for the calm/leisurely meaning. The reduplication 々 emphasizes the adjectival/adverbial quality.