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Please enjoy a relaxing time in our salon's calm atmosphere.
Literal
Settled atmosphere [possessive-の] salon [at-で] leisurely [adverbial-と] did time [object-を] please-enjoy.
A polished service-industry sentence. 落ち着いた雰囲気 ('calm atmosphere') uses the past participle of 落ち着く as a modifier. ゆったりとした is the mimetic ゆったり ('roomy, relaxed') in attributive form. お楽しみください is the keigo request form (お~ください). The whole sentence is crafted marketing language — warm but formulaic.