To languish is to become pitiful or weak because you're sick, in love, or stuck somewhere. A prisoner might languish in jail, longing for her freedom.
Languish, like languid, is from the Latin word languere which means to "be weak or faint." Your houseplants might languish in a dark dry corner. A Romantic poet might languish on a velvet couch with the back of her hand to her forehead. People in operas love to languish: The main character in La Traviatta, Violetta, languishes from longing and eventually tuberculosis.