It is well-known that étale cohomology is used in the proof of Weil conjectures and that SGA 4.5 is devoted to it. Also it seems(from a brief perusal of Milne's notes) that it is a kind of Galois Cohomology/Kummer theory for arbitrary algebraic varieties.
However I have heard a lot of people praising it, and this leads me to suspect that it must have applications beyond proving the Weil conjectures. I would be grateful if some of these can be given. I am sorry if this is a stupid question. The wikipedia page, Milne's article, etc., did not give any extra applications and so I hope asking people is more sensible. Please provide references also if available.
Best Answer
One of the most important applications of etale cohomology is to Deligne-Lusztig theory, and the large subsequent body of work approaching the representation of finite groups of Lie type using $\ell$-adic cohomology. For me, this is the most important application beyond the Weil conjectures.
In addition to the original paper of Deligne and Lusztig "Representations of reductive groups over finite fields" in Ann. of Math 1976, you might be interested in the book "Weil Conjectures, Perverse Sheaves and l'adic Fourier Transform", by Weissauer and Kiehl.