The differences of these three types of writing are easier to understand than perhaps their similarities.
Drama is often written in the form of a play. It is divided into acts, which are further subdivided into scenes. Drama can be comedic or tragic, but only includes the dialogue the characters are saying, sometimes accompanied by stage directions, which indicate what the characters are doing. Some people consider drama or plays difficult to read because the reader needs to use his/her imagine to find out what's going on, rather than the author's descriptions. Drama is alos meant to be acted out or read out loud, rather than read on the page. SOme examples of drama are any of Shakespeare's plays, Oedipus Rex and Antigone by Sophocles, The Importance of Being Earnest by Oscar Wilde, The Skin of Our Teeth by Thornotn Wilde, and The Glass Menagerie by Tennesee Williams.
Poetry stands out in its form--that is, the way it's written. Poetry, in the technical definition is writing with meter (or some kind of rhythm of stressed or unstressed syllables.) Often times, when you read a poem, the lines break mid-sentence and don't go all the way to the end of the page. That's because the author is counting the number of syllable and making the rhytm of the words fit his poetry (this is less often true in more contemporary poetry.) However, it is the sound of the words when read outloud that gives poetry its meaning. Authors do their best to condense as much meaning into a few words as they can. Poetry will often be filled with all kinds of figures of speech--similies, metaphors, personification, irony, etc--rather than simply stating a point and moving on. Some examples of poems include "Annabel Lee" by Edgar Allen Poe, "Because I Could Not Stop for Death--" by Emily Dickinson, any of Shakespeare's sonnets (and many portions of his plays), and "The Road Less Travelled" by Robert Frost. In addition, a lot of people consider song lyrics poetry.
Short stories are written in prose, which is the opposite of poetry--it's normal, every day lanuage that we speak. In a short story, an author tells a story that starts as close to the end as possible. Short stories usually only have one plot, a few characters and can be read in a relatively short period of time. They provide the read with brief descriptions of characters and places, and strive to tell a full story. Some examples include "The Black Cat" by Edgar Allen Poe, "The Gift of the Magi" by O. Henry, "The Lie" by Kurt Vonnegut, and "A Perfect Day for Bananfish" by J.D Salinger.
The similarities are that they are all usually works of fiction that strive to tell a complete story of the human experience. The form in which each genre accomplishes that is what makes them different.