Someone who is meticulous pays extreme attention to detail. If that person is, say, your surgeon or your accountant, you'll definitely want them to be meticulous!
The Latin root of meticulous is metus, which means "fear." Someone who's meticulous is afraid of what will happen if they're not careful enough to get every detail right. "Detail-oriented" and "perfectionist" are other ways of describing someone who cares deeply about the small things and about getting things exactly right, every time. Concert pianists must be meticulous, because audiences are always listening for wrong notes.