The Patronus Charm is a charm that evokes a partially-tangible, positive energy force known as a Patronus (pl. Patronuses Patronuses are also called spirit guardians though this may only refer to corporeal Patronuses. It is primarily designed for defence against otherwise unbeatable Dark creatures like Dementors and Lethifolds, though there are other uses such as messaging. It is apparent from old woodcuts and scrolls that the Patronus Charm has been used since ancient times.
To successfully cast the spell one must muster a happy memory (the happier the memory, the more powerful and tangible the Patronus will be) and incant "Expecto Patronum". It is unknown if there are any other elements required in casting, such as physical gestures (aside from drawing and pointing the wand forward prior to or during incantation). The happy memory one requires when casting does not necessarily have to be a specific moment in your past. Arguably more important than content, is how the memory or thought makes you feel now.
Harry Potter, for instance, is often seen to produce his more powerful patronuses when he his merely thinking of the people he loves. Harry also often thinks of his friends Ronald Weasley and Hermione Granger and produces successful Patronuses. One time Harry also used the thought of Dolores Umbridge being sacked to produce a patronus for his Defence Against the Dark Arts O. W. L. Harry uses his love and happiness, evoked from thinking of his friends and family, to produce a patronus, rather than having to remember emotions felt in the past.
A non-corporeal Patronus is one which does not resemble any living creature and has few distinguishing features (if any). Non-corporeal Patronuses resemble a burst of vapour or smoke without any clearly defined form shooting from the tip of the wand. While they may be partially effective at halting dementors (but not repelling them), non-corporeal Patronuses are not "fully-fledged" Patronuses, and are regarded as a more primitive or weaker version of the true Charm.