I am working on a small project and the project seems to require some very basic representation theory, in particular, representation of Clifford algebras. Since I have no experience whatsoever with representations, naturally I did what any other aspiring mathematician would do and turned to, of course, wikipedia.
According to wikipedia, a representation of an algebra $A$ on a $\mathbb{k}$-vector space $V$ is a map $\Phi : A \times V \rightarrow V$ such that
(i) for any $a \in A$, the map $\Phi(a):V \rightarrow V;~v \mapsto \Phi(a,v)$, is $\mathbb{k}$-linear.
(ii) If we denote $\Phi(a,v) = a \cdot v$, then for $a_1,a_2 \in A$ and $v \in V$ we have $e \cdot v = v$ and $a_1\cdot (a_2 \cdot v)=(a_1a_2)\cdot v$, where $e$ is the identity of $A$.
Note that we use $C_k$ as notation for the $k$'th Clifford algebra.
In the paper I am reading, they make the following statement:
Suppose $V$ is an $n$-dimensional vector space with a $C_k$-module structure $C_k \otimes_\mathbb{R} V \rightarrow V$. Then $V$ is a representation of $C_k$.
Here are my questions:
I am confused about what a " $C_k$-module structure $C_k \otimes_\mathbb{R}V \rightarrow V$ " means.
Am I using the proper definition of a representation, and if so, why is $V$ then a representation of $C_k$?
The two notions you outlined are equivalent. I'll outline the proof and leave the details to you, don't hesitate to ask if anything is unclear.
Let's suppose that al vector spaces are over a field $\mathbb{K}$. Let $A$ be an algebra, if you want you may assume that it's a Clifford algebra, but that's inessential. For simplicity of notation I will omit the subscript $\mathbb{K}$ from the tensor products.
First let's elucidate what an $A$-module structure on $V$ is. As in your statement, it is first of all a linear map \begin{equation} \Psi:A \otimes V \rightarrow V, \end{equation} with the property that $\Psi(e \otimes v) = v$. Second, note that there is a natural map \begin{equation} \mu:A \otimes A \otimes V \rightarrow A \otimes V, \end{equation} given by the extension of the map \begin{equation} (a,a',v) \mapsto (aa',v). \end{equation} If we want an $A$-module structure on $V$ we now demand that the following holds \begin{equation} \Psi \circ \mu = \Psi \circ (1 \otimes \Psi). \end{equation} Let's call this condition (I). This condition might seem complicated, but it's basically point (ii) in your description.
If the pair $(V, \Phi)$ is a representation of $A$ in the sense of wikipedia, then one obtains a map \begin{equation} A \otimes V \rightarrow V, \end{equation} by the universal property of the tensor product, since the map $A \times V \rightarrow V, (a,v) \mapsto \Phi(a) v$ is $\mathbb{K}$-bilinear.$^{1}$ One may check that the map obtained in this way satisfies condition (I), (you will need (ii)). So a representation in the sense of wikipedia implies a representation in the sense of Haynes.
In the other direction, if one has a representation in the sense of Haynes, that is a map $\Psi: A \otimes V \rightarrow V$ that satisfies property (I), then one obtains a representation in the sense of wikipedia by simply composing $\Psi$ with the natural map $A \times V \rightarrow A \otimes V$.
(1) This is a bit stronger than what you quoted from wikipedia, I'm not sure what article you found on wikipedia, but the condition that I wrote down is definitely the right one.