In this question, Eric Zaslow asks about $E_\infty$ algebras. Here I ask the same question but for $E_n$ algebras. Can anyone give me a plain-and-simple definition of an $E_n$ algebra, which avoids the word "operad"? (A definition that does involve the word "operad" is: An $E_n$ algebra is an algebra over (chains on) the little $n$-discs operad.) For $E_1$ algebras, which are the same as $A_\infty$ algebras, we have a definition involving a bunch of maps $m_n$ satisfying some quadratic relations. Is there any analogous definition for $E_n$? I'd be interested in even just $E_2$ or $E_3$.
Edit: I have nothing against operads. As Charles Rezk points out, I'm just asking for a nice presentation of an $E_n$ operad which yields a nice definition of $E_n$ algebra involving some nice generators and some nice relations. I realize that "the reason" for the nice definition of $A_\infty$ algebra is because of the Stasheff associahedra. So I wonder if there are any analogous things for $E_n$? Or, if such a nice presentation is not possible or not feasible, then why not?
(Assume characteristic zero if needed.)