Satire has changed its meaning during the twentieth century. Academics, particularly in Arts subjects, prefer to use the traditional - and more precise - meaning. This is "writing which is meant to correct the errors or vices of people by holding them up to mockery or scorn". It is, in this traditional sense, a serious piece of writing, with the purpose of changing - improving - the writer's society, either in the general social or the political way. Nowadays it is more loosely applied to writing that is funny in an approximately critical direction. Traditionally, satire was not always funny.
Traditional satire often uses allegory as its tool. (Two examples of allegory given on that page are also good examples of satire - Animal Farm and Gulliver's Travels). Figures of speech can often be used together. Irony is another tool used by satirists; and sarcasm can be confused with it.
The adjective is satirical when you are trying to communicate the tone of a piece of writing; and satiric when it is used as a rather more technical description in literary studies. The adverb from both is satirically.