Nastily Exhausting Wizarding Test

A Nastily Exhausting Wizarding Tests (often abbreviated N.E.W.T.) is a subject-specific exam that seventh year wizards and witches at Hogwarts School of Witchcraft and Wizardry take to help them pursue certain careers after their graduation. For instance, the Ministry of Magic only accepts Auror applicants with at least five N.E.W.T.s with the grade "Exceeds Expectations" or better.

A student must have previously achieved an O.W.L. in the subject in order to move on to N.E.W.T.-level classes. Most Hogwarts Professors will accept an "Exceeds Expectations" in the student's O.W.L. for a class in order to allow them into the N.E.W.T.-level classes. However, some, such as Severus Snape, would only let students with an "Outstanding" in their O.W.L. pass through to his Potions N.E.W.T. class, although he did accept students with "Exceeds Expectations" into his N.E.W.T.-level D.A.D.A. class during the 1996–1997 school year.

Aurors

 * Alastor Moody
 * Alice Longbottom
 * Frank Longbottom
 * Gawain Robards
 * John Dawlish
 * Kingsley Shacklebolt
 * Nymphadora Tonks
 * Proudfoot
 * Rufus Scrimgeour
 * Savage
 * Williamson

Healers

 * Dilys Derwent
 * Lancelot
 * Augustus Pye
 * Hippocrates Smethwyck
 * Helbert Spleen
 * Miriam Strout

Others

 * The Marauders and Lily Evans
 * Albus Dumbledore
 * Minerva McGonagall (earned top grades in her N.E.W.T.s)
 * Barty Crouch Jr.
 * Percy Weasley
 * Tom Riddle
 * Hermione Granger (returned after Battle of Hogwarts to take N.E.W.T.s)
 * Most likely others who worked for the Ministry of Magic, such as Arthur Weasley, Mafalda Hopkirk, Amelia Bones, and Dolores Umbridge.
 * Hogwarts Professors most likely have taken N.E.W.T.s, at least in their own subject.

Notable students who did not pass N.E.W.T.S.
Not all wizards pass or even take the N.E.W.T.s, as they can find other jobs using their O.W.L. results. Fred and George Weasley, for example, were not bothered about taking the N.E.W.T.s and left Hogwarts in the middle of school-term (since their ambition was to open their joke shop, results were not an imperative in their opinion). However, it should be noted that the immediate cause of their departure was to evade reprisals from the much-disliked Professor Umbridge for the spectacular prank they had pulled on her earlier that day.
 * Harry Potter (didn't return to school for seventh year; went to work for the Ministry as an Auror, ends up becoming the Head of Auror Office in 2007).
 * Ron Weasley (didn't return to school for seventh year; went to work at the joke shop with his brother George, later worked at the Ministry as an Auror).
 * Neville Longbottom (went to work for the Ministry as an Auror, then returned to Hogwarts as a professor in Herbology).
 * Fred and George Weasley (mentioned above, left during the rule of Dolores Umbridge, before exams were over and started their own joke shop).

Behind the scenes

 * The N.E.W.T. is comparable to the Muggle A-Levels given at the same stage of education in most of the UK and Commonwealth,the leaving certificate given in Ireland, the Advanced Placement exams given in the United States, the Higher School Certificate (or comparable certificates) given in Australia, or the International Baccalaureate Diploma Programme given worldwide.
 * Harry Potter, Ron Weasley, and Hermione Granger skipped the 1997–1998 school year, which would have been their final year, due to the reign of Lord Voldemort and their search for his Horcruxes. Presumably, Muggle-born students also missed their schooling that year. After the defeat of Voldemort, Hermione completed her education and took her N.E.W.T.s, while Harry became an Auror and Ron helped his brother George at Weasleys' Wizard Wheezes. Those who participated in the Battle of Hogwarts were given exemptions from usual Auror qualifications by Kingsley Shacklebolt due to their duelling experience.
 * Doing well on N.E.W.T.s is normally imperative to earning a position at the Ministry of Magic, except under special circumstances (see above).
 * N.E.W.T.s are the highest level of examinations given at Hogwarts. It is unknown what system other wizarding schools such as Beauxbatons or Durmstrang use.

Notes and references
A.S.P.I.C.