What is a cusp parameter? I was reading this paper, and on the first page they define a cusp form as
$$
f(z) = \sum_{n > -\alpha} a(n) e^{2\pi i (n + \alpha)z}.
$$
Is this equivalent to the usual definition of a cusp form
$$
f(z) = \sum_{n = 1}^\infty a(n) q^n.
$$
where $q = e^{2\pi iz}$? 
Also what is a cusp parameter?
 A: Short answer:
When you translate a cusp to $\infty$, the cusp parameter measures the default of periodicity of a modular (or Maass) form under the group $\begin{pmatrix}1 & * \\ 0 & 1\end{pmatrix}$. It can be nonzero only when the cusp is not already $\infty$, and when the form has nontrivial nebentypus.
Definition:
You can find a definition in Goldfeld, Hundley, Automorphic representations and L-functions for the general linear group, Definition 3.7.3:
Let $N > 0$ be an integer and $\mathfrak a \in \mathbb P^1(\mathbb Q)$ a cusp for $\Gamma_0(N)$. There exists a unique $g$ in the stabilizer of $\mathfrak a$ in $\Gamma_0(N)$ that is conjugate in $\operatorname{SL}(2, \mathbb R)$ to $\begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}$. Write
$$g = \begin{pmatrix}a & b \\ c & d\end{pmatrix} \in \Gamma_0(N) \,.$$
Let $\chi$ be a Dirichlet character modulo $N$. The cusp parameter of $\mathfrak a$ w.r.t. $\chi$ is the unique real number $\mu\in[0,1)$ with $\exp(2\pi i \mu) = \chi(d)$.

Fourier series:
There exists a unique $\sigma \in \operatorname{SL}(2, \mathbb R)/\operatorname{Stab}_{\operatorname{SL}(2, \mathbb R)}(\infty)$ with
$$\sigma \infty = \mathfrak a, \qquad \sigma^{-1} g \sigma = \begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix} \,.$$
Let $f$ be a modular (or Maass) form of weight $k$, level $N$, nebentypus $\chi$. Consider $(f|_k \sigma )(z) = j(\sigma, z)^{-k} f(\sigma z)$. Slashing this by $\begin{pmatrix}1 & 1 \\ 0 & 1\end{pmatrix}$ amounts to slashing $f$ by $g$. We thus see that
$$(f|_k \sigma )(z+n) = \chi(d)^n (f|_k \sigma )(z) = e^{2 \pi i n \mu} (f|_k \sigma )(z)$$
for $n \in \mathbb Z$. This means that
$$e^{-2 \pi i \mu z} (f|_k \sigma )(z)$$
is $\mathbb Z$-invariant, and has a Fourier expansion of the form $\sum_{n \geq 0} a_n q^n$ (if $f$ is modular, say).
Cusp forms: $f$ is cuspidal iff "its constant term" at every cusp vanishes. When $\mathfrak a$ is a cusp with cusp parameter $0$, this means that $a_0 = 0$ for this cusp. When its cuspidal parameter is nonzero, the Fourier expansion of $f$ has no constant term, and there is no condition on $a_0$. (In this case, $f|_k \sigma$ decays exponentially at $\infty$, even though $e^{-2 \pi i \mu z} (f|_k \sigma )(z)$ may have a nonzero limit at $\infty$.)
This corresponds to what the authors define as a cusp form: when $\alpha=0$ they require $a_0 =0$, and when $\alpha > 0$ there is no condition on $a_0$.
This definition of a cusp form may seem a bit unnatural, but it is very natural from an adelic point of view, where it really corresponds to the vanishing of the constant term in the adelic Fourier expansion.
