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.
Best Answer
Ends and coends should be thought of as very canonical constructions: as Finn said, they can be described as weighted limits and colimits, where the weights are hom-functors.
Recall that if $J$ is a (small) category, a weight on $J$ is a functor $W: J \to Set$. The limit of a functor $F: J \to C$ with respect to a weight $W$ is an object $lim_J F$ of $C$ that represents the functor
$$C^{op} \to Set: c \mapsto Nat(W, \hom_C(c, F-)).$$
Dually, given a weight $W: J^{op} \to Set$, the weighted colimit of $F: J \to C$ with respect to $W$ is an object $colim_J F$ that represents the functor
$$C \to Set: c \mapsto Nat(W, \hom_C(F-, c)).$$
Then, as Finn notes above, the end of a functor $F: J^{op} \times J \to C$ is the weighted limit of $F$ with respect to the weight $\hom_J: J^{op} \times J \to Set$, and the coend is the weighted colimit of $F$ with respect to $\hom_{J^{op}}: J \times J^{op} \to Set$.
The ordinary limit of $F$ is the weighted limit of $F$ with respect to the terminal functor $t: J \to Set$. Ordinary limits suffice for ordinary ($Set$-based) categories, but they are inadequate for enriched category theory. The concept of weight was introduced to give an adequate theory of enriched limits and colimits (replacing $Set$ by suitable $V$, and functors as above by enriched functors, etc.)
Weighted colimits and weighted limits (in particular coends and ends) can be expressed in terms of Kan extensions. For any weight $W$ in $Set^{J^{op}}$, the weighted colimit of $F: J \to C$ (if it exists) is the value of the left Kan extension of $F: J \to C$ along the Yoneda embedding $y: J \to Set^{J^{op}}$ when evaluated at $W$, in other words
$$(Lan_y F)(W)$$
A similar statement can be made for weighted limits, as values of a right Kan extension.