Reddit reviews Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (Ergebnisse der Mathematik und ihrer ... Series of Modern Surveys in Mathematics (34))

The Gagliargo-Nirenberg inequalities you mention originate here.

Some of Nirenberg's "greatest hits" at a glance: some of his early work concerned the Minkowski problem of finding surfaces with prescribed Gauss curvature, and the related Weyl problem of finding isometric embeddings of positive curvature metrics on the sphere. For a gentle introduction to this type of problem accessible (in principle) after a basic course in differential geometry and some analysis, see these notes by Khazdan. For a more advanced treatment, including a discussion of the Minkowski problem and generalizations see these notes by Guan. This line of research owes a lot to Nirenberg.

In this legendary paper (2700+ citations, for a math paper!) and another with the same co-authors (Agmon and Douglis), he investigated boundary Schauder and L^p estimates for solutions of general linear elliptic equations. You can look at Gilbarg-Trudinger, or Krylov's books (1, 2) for the basics of linear elliptic equations, including boundary estimates. Here is a course by Viaclovsky in case you don't want to buy the books. This last set is far more basic stuff than Agmon-Douglis-Nirenberg, though, but it should give you an idea of what its about.

Another extremely famous contribution of Nirenberg is his introduction with Kohn of the (Kohn-Nirenberg) calculus of pseudodifferential operators. Shortly thereafter, Hoermander began his monumental study of the subject, later summarized in his books I, II, III, IV. If you know nothing about pseudo-differential operators, I suggest starting with this book by Alinhac and Gérard.

Another gigantic result is the Newlander-Nirenberg theorem on integrability of almost complex structures. An almost-complex structure is a structure on the tangent space of a manifold which mimics the effect that rotation by i has on the tangent vectors. The Newlander-Nirenberg tells you that if a certain simple necessary condition holds, you can actually choose locally holomorphic coordinates for the manifold compatible which induce this a.c. structure. A proof that should be reasonably accessible, provided you understand what I just wrote and have some basic notions of several complex variables can be found here.

Nirenberg also studied the important problem of (local) solvability of (pseudo)-differential equations with Francois Treves. In this paper, he introduced the famous condition Psi, which was only recently proved by Dencker to be necessary and sufficient for local solvability. An exposition of the problem at a basic level can be found in this undergrad thesis from UW.

Another massively influential paper was this one, with Fritz John, where he introduces the space of BMO functions, and proved the Nirenberg-John lemma to the effect that any BMO function is exponentially integrable. Fefferman later identified BMO as the dual of the Hardy space Re H_1, and the BMO class plays a crucial role in the Calderon-Zygmund theory of singular integral operators. You can read about this in any decent book on harmonic analysis. I myself like Duoandicoetxea's Fourier Analysis. BMO functions are treated in chapter 6. For a more "old school" treatment using complex analysis, including a proof of Fefferman's theorem, check out Koosis' lovely Introduction to H^p spaces.

Another noted contribution was his "abstract Cauchy-Kowalevski" theorem, where he formulated the classical theorem in terms of an iteration in a scale of spaces, instead of the more direct treatment based on power series. This point of view has now become classical. Look at the proof in Treve's book Basic Linear Partial Differential Equations.

Next, his landmark paper with Gidas and Ni (2000+ citations) on symmetry of positive solutions to second order nonlinear elliptic PDE are absolute classics. The technique is now a basic part of the "elliptic toolbox".

His series of papers with Caffarelli, Spruck and Kohn (starting here) on fully nonlinear equations is also classic, and the basis for much of the later work. It's gotten sustained attention in part because optimal transport equations are of (real) Monge-Ampere type.

The theorem about partial regularity of NS you are referring to is this absolute classic with Cafarelli and Kohn. A simple recent proof, together with an accessible exposition of de Georgi's method, can be found here.

Let me finish by mentioning my personal favorite, one of the most cited papers in analysis of the 20th century, an absolute landmark of variational analysis, Brezis-Nirenberg 1983. A pedagogical exposition appears in Chapter III of Struwe's excellent book.

TLDR: Nirenberg is one of the most important analysts of the past 60 years.

edit: Thanks for the gold! Glad this was useful/interesting to someone, given how advanced and specialized the material is.