This is somehow a general (and naive) question, but as specialized mathematicians we usually miss important results outside our area of research.
So, generally speaking, which have been important breakthroughs in 2021 in different mathematical disciplines?
Advancing mathematics by guiding human intuition with AI, Nature 600, 70 (2021), stands out because it represents the first significant advance in pure mathematics generated by artificial intelligence.
More newsworthy items (each item has a link to a blog on Quanta magazine for an informal discussion of its significance):
Strictly speaking this is not a new mathematical result (meaning no new proof), but let me mention the Liquid Tensor Experiment, the verification in Lean of a very recent theorem by Clausen and Scholze.
Here is the original post by Scholze, here the story six months later, the canonical quanta link and, last but not least, a nature article.
PS: I participated in the project, so my opinion about its importance is surely biased.
One of the most exciting developments in combinatorics in 2021 is the proof of the Erdos-Faber-Lovasz Conjecture on the chromatic index of hypergraphs. There is a good article in Quanta magazine about the proof.
In (analytic) number theory Paul Nelson's recent preprint (https://arxiv.org/abs/2109.15230) solved the subconvexity problem for a huge class of L-functions in the t-aspect.
More precisely subconvexity bounds for $L(\frac{1}{2}+it, \pi, St)$ are established for cuspidal automorphic representations of $GL_n$.
This is a huge breakthrough and also the methods are very exciting and promising.
Edit: Now there is an article on this result on Quanta magazine.
My favourite theorems in mathematics are the ones that at the same time great and have easy-to-understand formulation. To put aside various P=NP claims in arxiv, I will concentrate on theorems that where peer-reviewed and published in 2021. Most of them appeared in arxiv before.
So, the greatest easy-to-understand theorems published in 2021 are:
Theorem of Annika Heckel about non-concentration of the chromatic number of a random graph https://www.ams.org/journals/jams/2021-34-01/S0894-0347-2020-00957-2/
A dramatic progress on sunflower conjecture by Alweiss, Lovett, Wu and Zhang https://annals.math.princeton.edu/2021/194-3/p05
The resolution of the rectangular peg problem for smooth Jordan curves by Greene and Lobb https://annals.math.princeton.edu/2021/194-2/p04
Characterization of fields of values of odd-degree irreducible characters by Navarro and Tiep https://www.cambridge.org/core/journals/forum-of-mathematics-pi/article/fields-of-values-of-characters-of-degree-not-divisible-by-p/0FE327DAE6088EC9B43A0CFAFC8E8D52
Proof that the Dehn function of $\mathrm{SL}_4({\mathbb Z})$ is quadratic by Leuzinger and Young https://annals.math.princeton.edu/2021/193-3/p02
Proof that the group $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for all $n \geq 6$ by Kaluba, Kielak and Nowak https://annals.math.princeton.edu/2021/193-2/p03
Characterization of $\frac{3}{2}$-generated groups by Burness, Guralnick and Harper https://annals.math.princeton.edu/2021/193-2/p05
A counterexample to the unit conjecture for group rings by Gardam https://annals.math.princeton.edu/2021/194-3/p09
Proof of the universal optimality of the ${E}_8$ and Leech lattices by Cohn, Kumar, Miller, Radchenko and Viazovska https://annals.math.princeton.edu/articles/17703
New bounds on the density of lattice coverings by Ordentlich, Regev and Weiss https://www.ams.org/journals/jams/2022-35-01/S0894-0347-2021-00984-0/home.html
Determining, for each given fixed angle and in all sufficiently large dimensions, the maximum number of lines pairwise separated by the given angle by Jiang, Tidor, Yao, Zhang and Zhao https://annals.math.princeton.edu/2021/194-3/p03
The isoperimetric inequality for minimal surfaces by Brendle https://www.ams.org/journals/jams/2021-34-02/S0894-0347-2021-00969-4/
A connection between thresholds and fractional expectation-thresholds established by Frankston, Kahn, Narayanan and Park https://annals.math.princeton.edu/2021/194-2/p02
An almost constant lower bound of the isoperimetric coefficient in the KLS conjecture by Chen https://link.springer.com/article/10.1007/s00039-021-00558-4
An $O(n \log n)$ algorithm for multiplication of $n$-digit integers by Harvey and van der Hoeven https://annals.math.princeton.edu/2021/193-2/p04
An explicit construction of polynomials with optimal condition number by Beltran, Etayo, Marzo and Ortega-Cerd https://www.ams.org/journals/jams/2021-34-01/S0894-0347-2020-00956-0/
I am sorry if you think that this list is too long but in my opinion all these theorems are both great and beautiful, so I will let you to choose your own 3-5 favourite ones.
Finally, you may want to look at my book https://link.springer.com/book/10.1007/978-3-030-80627-9 with the descriptions of all such theorems published from 2001 until now.
Having just listened to some of Jacob Tsimerman's Minerva lectures, I became aware of the recent arXiv preprint, Canonical Heights on Shimura Varieties and the André–Oort Conjecture, by Jonathan Pila, Ananth N. Shankar, Jacob Tsimerman, Hélène Esnault, and Michael Groechenig. Assuming the paper is correct, it gives the first unconditional (i.e., not assuming the Generalized Riemann Hypothesis) proof of the full André–Oort Conjecture. The proof builds on a lot of previous work and knits together a wide variety of techniques and ideas, but one thing that I find personally appealing is that the theory of o-minimality plays a key role behind the scenes. A priori, one might not guess that model theory has much to say about counting rational points, but it does!
Dmitri Pavlov and Daniel Grady released a preprint containing the first complete proof of the Cobordism Hypothesis, and in fact they prove a significant generalization to cobordism categories with geometric structure. Their article has a good discussion of prior work on this problem.
Since other answers mention works published in 2021, I think one can add to the list the proof of triviality of the $\phi^4$ quantum field theory in four dimensions:
Michael Aizenman, Hugo Duminil-Copin, "Marginal triviality of the scaling limits of critical 4D Ising and $\lambda\phi_4^4$ models", Ann. of Math. (2) 194(1): 163-235 (July 2021).
The negative answer by counterexample to the Modular Isomorphism Problem for group rings (that is, the question whether for $p$-groups $G$ and $H$, the group rings ${\mathbb F}_p G$ and ${\mathbb F}_p H$ are isomorphic only if $G$ and $H$ are isomorphic) by Garcia-Lucas, Margolis and del Rio.
García-Lucas, Diego; Margolis, Leo; del Río, Ángel, Non-isomorphic 2-groups with isomorphic modular group algebras, J. Reine Angew. Math. 783, 269-274 (2022). ZBL1514.20019.