An $E_n$ algebra (an algebra over the little $n$-cubes operad, etc.) is intuitively an object with $n$ compatible monoid structures. All of the subtlety in this theory lies in making "compatible" precise; in particular it is not a property but a structure.

Here are some examples.

- In $\text{Set}$, an $E_1$ algebra is a monoid. For $n \ge 2$ an $E_n$ algebra is a commutative monoid by the Eckmann-Hilton argument.
- In $\text{Ab}$, an $E_1$ algebra is a ring. For $n \ge 2$ an $E_n$ algebra is a commutative ring by the Eckmann-Hilton argument.
- In $\text{Top}$, a grouplike $E_n$ algebra (an $E_n$ algebra where $\pi_0$ is a group) is an $n$-fold loop space $\Omega^n X$ by the recognition principle. The $n$ monoid structures are the loop compositions in the $n$ loop directions. A grouplike $E_{\infty}$ algebra is an infinite loop space, or equivalently a connective spectrum.
- In $\text{Cat}$, an $E_1$ algebra is a monoidal category, an $E_2$ algebra is a braided monoidal category, and for $n \ge 3$ an $E_n$ algebra is a symmetric monoidal category. (This stabilization phenomenon is related to Freudenthal suspension and is part of the "periodic table of higher categories.")

In more detail, let's focus on $n = 2$. An $E_2$ algebra is intuitively an object with two compatible monoid structures. The Eckmann-Hilton argument shows that they're equivalent, but the way in which it shows that they're equivalent is itself interesting structure: along the way, it describes a map between $ab$ and $ba$ (for both monoidal structures), which is a braiding. This is where braided monoidal categories come into the picture. It might help to stare at the standard proof of Eckmann-Hilton where you move squares around and to explicitly think of the squares as describing binary operations in the little $2$-cubes operad.

One way to make "compatible" precise is to write down a presentation of the $E_2$ operad, which is unique among the $E_n$ operads ($n \ge 2$) in that all of its spaces are $1$-truncated: that is, they are all groupoids, or equivalently have no higher homotopy $\pi_n, n \ge 2$. There is a particularly nice model of the $E_2$ operad as an operad in groupoids called the parenthesized braid operad, which has a "generators-and-relations" presentation, but where it's important to understand that when specifying an operad in groupoids there are three sorts of things you might want to write down, rather than two:

- Operations (which then generate other operations under operadic composition),
- $1$-morphisms between operations (to describe the groupoid structure), and
- Relations between $1$-morphisms between operations (to further describe the groupoid structure).

In general, higher category theory blurs the distinction between generators and relations: relations become generators one categorical level up. To avoid having to make the distinction you can just say "presentation."

The standard presentation of the parenthesized braid operad mimics exactly the standard axiomatization of braided monoidal categories: there is a generating binary operation (the monoidal structure), two generating $1$-morphisms (the associator and the braiding), and some relations between these (the pentagon and hexagon axioms). Here I'm ignoring units for simplicity. This presentation is important in discussions of Grothendieck-Teichmüller theory.

For $n \ge 3$ the spaces in the $E_n$ operads aren't truncated: for example the space of binary operations is the sphere $S^{n-1}$, which has nontrivial homotopy groups in arbitrarily high degrees. So I don't think there's any hope for a presentation along the above lines in general. You could ask for presentations of various truncations, but I imagine these are pretty horrible to work with in general. The $E_n$ operads exist precisely so that you can avoid having to do stuff like this. Of course it's a different story after taking rational homology.

$E_n$ algebras show up in the story of factorization homology and topological field theory, so that's one place to go for some resources; see, for example, these notes by Ginot.