Drama usually indicates a piece of writing that is intended to be acted out - a performance on a stage of some sort with people acting as the characters in the story, allowing an audience to see the action unfold.
Poetry is frequently written with a particular awareness of the sound of the language being used. Poetry may or may not rhyme but the rhythm of the language will be of central importance and the author may have spent extra time working to achieve carefully crafted, vivid word pictures.
Short stories are just that - stories that are of a length that would ideally allow them to be read from start to finish in one sitting.
These can overlap - poetry can be presented as a dramatic play; a short story may be written in poetic form, etc. The three formats are different but not exclusive types of literature, presenting stories in three different formats.
poetry is language artfully arranged using tools of sound, rhythm, shape of lines, image and metaphor to create an emotional response in the reader. David St.John says poetry creates empathy with that which is other. Poems can be narrative (tell a story) and well-wriiten prose can be poetry.
A short story is a work of fiction that is usually written in prose, often in narrative format. This format tends to be more pointed than longer works of fiction, such as novellas (in the 20th and 21st century sense) and novels.
Drama is the specific mode of fiction represented in performance.A prose or verse composition, especially one telling a serious story, that is intended for representation by actors impersonating the characters and performing the dialogue and action.