All of these literary forms are used to tell a story of some kind, but all three use different structural forms in order to do so.
Poetry is usually written in verse and structured by separate lines (and often stanzas) rather than in tradtional prose/paragraph form. It is often written for aesthetic purposes to arouse the emotions, and poetry utilizes literary devices such as meter and rhythm often not found in other forms. Other literary devices such as assonance, metaphors and alliteration are commonly found. Poems can be of a rhyming or non-rhyming nature.
Short stories are fictional pieces, usually written in a narrative format of 30 pages or less (as compared to the longer novella and novel), and written in prose, with sentences and paragraphs. They generally describe a much shorter event than the longer prose forms.
Drama can occur in poems, short stories, novellas and novels, but it is generally regarded as (or confused with) the Play--the type of distinctive structural form used in theatre. It always includes some form of action and conflict. Plays have their own specific structure, written in scenes and/or acts, with dialogue representing the predominant narrative. The dialogue of plays are written for individual characters, and are noted as such within the script. Stage directions and other non-dialogue instructions can also be included.