The following is really an adjunction between $2$-categories but I am going to ignore that subtlety. This blog post discusses everything in more detail and with a few more examples.

Consider on the one hand all concrete categories, by which I mean pairs $(C, U)$ of a category $C$ and a functor $U : C \to \text{Set}$ (the "underlying set") functor, and on the other hand the inclusion of those concrete categories which arise as the categories of models of a Lawvere theory $T$. Here by a Lawvere theory I mean for simplicity a category with finite products and objects $1, x, x^2, \dots$ for a distinguished object $x$, and by the category of models of a Lawvere theory I mean the category of product-preserving functors $T \to \text{Set}$, with the underlying set functor given by evaluation at $x$. Many familiar concrete categories of algebraic objects arise in this way, e.g. groups, rings, modules.

This inclusion has a left adjoint sending a concrete category $(C, U)$ to the "closest approximation" of that concrete category by the category of models of a Lawvere theory, which is the following. The full subcategory of the category of functors $C \to \text{Set}$ on the products $1, U, U^2, \dots$ of $U$ can be thought of as the category of operations on the objects of $C$ (e.g. natural transformations $U^n \to U$ correspond to the $n$-ary operations), and these naturally form a Lawvere theory which one can take the category of models of. Moreover, there is a natural functor of concrete categories from $(C, U)$ to the category of models of the Lawvere theory $T_U$ determined by $U$; this is the unit of the adjunction.

Alright, now for some examples in analysis, Lie theory, and differential geometry.

- Let $(C, U)$ be the concrete category of Banach spaces and weak contractions (maps of norm at most $1$), where $U$ sends a Banach space to its unit ball, and broaden the definition of a Lawvere theory to allow infinite products. Then the category of models of the infinitary Lawvere theory $T_U$ is the category of totally convex spaces.
- Let $(C, U)$ be the concrete category of commutative Banach algebras. Then the Lawvere theory $T_U$ is the Lawvere theory of holomorphic functions: it is equivalent to the category with objects $\mathbb{C}^n$ and morphisms holomorphic functions. This is a converse to the holomorphic functional calculus, and produces the holomorphic functional calculus itself from just the concrete category of commutative Banach algebras; in particular we did not have to know what a holomorphic function was in advance.
- Let $(C, U)$ be the concrete category of representations of a Lie algebra $\mathfrak{g}$. Then the category of models of the Lawvere theory $T_U$ is the category of representations of the universal enveloping algebra $U(\mathfrak{g})$. This is a fancy way of saying that on representations of $\mathfrak{g}$ one has more operations than just acting by elements of $\mathfrak{g}$: one can in fact act by elements of $U(\mathfrak{g})$. But it also says that all operations on representations of $\mathfrak{g}$ arise in this way.
- Let $(C, U)$ be the concrete category $\text{Man}^{op}$, the opposite of the category of finite-dimensional smooth manifolds, with $U$ given by sending a smooth manifold to its ring of smooth functions. Then the Lawvere theory $T_U$ is the Lawvere theory of smooth functions: it is equivalent to the category with objects $\mathbb{R}^n$ and morphisms smooth functions. The category of models of $T_U$ is the category of smooth algebras. The opposite of this category is the starting point of some approaches to synthetic differential geometry.