Here is a (non-exhaustive) list of cool results and techniques I particularly enjoyed learning about, in no particular order:
Moser's entropy compression argument, which I learned through Emo Welzl's fantastic course (see also here).
Kalai, Mitzenmacher, and Sudan's proof of the deletion channel capacity upper bound for small deletion probability.
Mitzenmacher and Drinea's (1-d)/9 lower bound on the deletion channel capacity.
Levenshtein's proof that Varshamov-Tenengolts codes correct a single deletion, and much more on Sloane's fun survey.
Dvir's proof of the finite field Kakeya conjecture. Read this great survey about its applications in pseudorandomness.
Marcus, Spielman, and Srivastasa's interlacing polynomials.
Guruswami, Umans, and Vadhan's almost optimal constructions of expanders and extractors.
Maurer's proof that secret-key agreement is possible in settings where the adversary is much stronger than the honest parties.
Knuth's balanced codes (short and sweet).
Krasikov and Roditty's approach to the k-deck problem.
Csirmaz's information-theoretic approach to share length lower bounds for secret sharing.
Dumer, Micciancio, and Sudan's proof that the minimum distance problem over linear codes is NP-hard to approximate via locally dense codes. In a related vein, Khot's proof that the shortest vector problem is NP-hard to approximate via locally dense lattices.
Rao's amazing walkthrough of Bourgain's two-source extractor.
De, O'Donnell, and Servedio's and Nazarov and Peres' complex analytic approach to the worst-case trace reconstruction problem.