What are differential algebras? On the bottom of page 10 of this paper (/!\ It's in french !) they talk about the 'differential $\mathbb{C}$-algebra of convergent series $\{C(x);x^2d/dx\}$' and about the 'differential $\mathbb{C}$-algebra of the germ of functions holomorphic at the origin $\{\mathcal{O}_0;x^2d/dx\}$'
What are those ?
 A: A differential algebra $A$ over a field $k$ is an algebra over $k$ equipped with a differential $d$; that is, a linear map $d: A \to A$ such that $d(fg) = fdg + gdf$. The name is motivated by the algebra of smooth functions $\Bbb R \to \Bbb R$ whose differential is, well, taking the derivative.\
Let's talk about your two algebras. First, we have the algebra of convegent power series, whose differential is $d=x^2\frac{d}{dx}$; explicitly, this means $$d: \sum_{n=0}^\infty a_nx^n \mapsto \sum_{n=2}^\infty (n-1)a_{n-1}x^n.$$ (We get this by first taking the derivative, then multiplying by $x^2$.) It's tedious but not hard to see that this satisfies the product rule $d(fg)=fdg+gdf$. 
Note that this is all over $\Bbb C$, and is taking place in the complex plane, rather than on the real line.
Talking about germs of functions is a bit more advanced, but once you understand what germs of holomorphic functions are, seeing that the latter algebra has the structure of a differential algebra is just a special case of seeing that the former does.
