So let $\Gamma\subseteq SL\_2(\mathbf{Z})$ be a finite index subgroup (not necessarily a congruence subgroup). Recall that we have an action of $SL\_2(\mathbf{R})$ on 
$\mathbb{H}=\{z\in\mathbf{C}:\Im(z)>0\}$ by moebius transformations and therefore of $\Gamma$. If $\tau\in\mathbb{H}$ and $[a,b,c,d]=\gamma\in SL\_2(\mathbf{Z})$ then we have an isomorphism
of complex tori
$$
\mathbf{C}/(\mathbf{Z}+\tau\mathbf{Z})\rightarrow \mathbf{C}/((a\tau+b)\mathbf{Z}+(c\tau+d))
\mathbf{Z}\rightarrow (\mathbf{C}/\mathbf{Z}+\gamma\tau\mathbf{Z}) \;\;\;\; (\*)
$$
where the first map is the identity and the second map is the multiplication by $(c\tau+d)^{-1}$. Let
$$
\tilde{\mathcal{E}}\_{\Gamma}=\{(\tau,x):\tau\in\mathbb{H},x\in \mathbf{C}/(\mathbf{Z}+\tau\mathbf{Z})\}
$$
We have a natural left action of $\Gamma$ on $\tilde{\mathcal{E}}\_{\Gamma}$ given by
$$
\gamma(\tau,x)=(\gamma\tau,j(\gamma,\tau)^{-1}x),
$$
which is just a reinterpretation of $(\*)$. Here $j(\gamma,\tau)=c\tau+d$. We thus get the following family of curves (note that the fibers are not necessarily elliptic curves because of the presence of torsion in $\Gamma$ as K. Buzzard pointed out):
$$
\pi\_\Gamma:\Gamma\backslash\tilde{\mathcal{E}}\_{\Gamma}=:\mathcal{E}\_{\Gamma}\rightarrow Y\_{\Gamma}:=\Gamma\backslash \mathbb{H}
$$
In the case where $\Gamma$ is torsion free, we readily see that the fibers are elliptic curves and the the $\Gamma$ action is compatible with the addition on the $tori$ (this somehow justifies the terminology "universal elliptic curve" over $Y\_{\Gamma}$).


In general one always has that $Y\_{\Gamma}$ is a quasi-projective curve defined over $\mathbf{C}$ (in fact it is always possible to define this curve over $\overline{\mathbf{Q}}$, the algebraic closure of $\mathbf{Q}$). 


[For example, when $\Gamma=\Gamma\_0(N)$ one may look at the modular polynomial $f\_N(x,y)\in\mathbf{Z}[x,y]$. Then using the modular interpretation one can show that there exists a (complex analytic) immersion of $Y\_{\Gamma\_0(N)}$ onto the plane curve $C\_N: f\_N(x,y)=0$. If I remember correctly, in the finite chart $\mathbf{C}^2$ (for $N$ large enough) the singularities of $C\_N$ are nodes and thus one can blow-up these points (over $\mathbf{Q}$!). From this one can construct a complex analytic isomorphism between $Y\_{\Gamma\_0(N)}$ and the blow-up (which is quasi-projective curve defined over $\mathbf{Q}$.]


So here are 2 natural questions.


**Q1**: Is $\mathcal{E}\_{\Gamma}$ quasi-projective (at least when $\Gamma$ is torsion free)?


**Q2**: If the answer is yes, then what is the **cleanest** (and if possible most transparent) way of showing that $\mathcal{E}\_{\Gamma}$ is quasi-projective? 


So for the second question, the thing that I have in mind would be to 1) construct some complex analytic immersion $\pi:\mathcal{E}\_{\Gamma}\rightarrow Z$, where $Z$ is
a quasi-projective surface and 2) performing a sequence of blow-ups on $Z$ I would try to construct an embedding. 


**added**: Note that one can always find a normal finite index subgroup $\Gamma'\leq \Gamma$.
Since $\mathcal{E}\_{\Gamma}=(\mathcal{E}\_{\Gamma'})^{\Gamma/\Gamma'}$ we readily see that if $\mathcal{E}\_{\Gamma'}$ is affine then automatically $\mathcal{E}\_{\Gamma}$ is affine being the quotient of affine variety by a finite group.



On the algberaicity of the universal elliptic curve associated to a torsion free subgroup
Here is an alternative take on Donu's argument: Removing the image $\sigma$ of a section of 
$E\_\Gamma$ allows one to regard a fiber of $E\_{\Gamma}$ as a once-punctured torus $S$. 
(In order to construct a section, use the standard upper half-space model of the Teichmuller space of the tori, so that each marked torus is identified with the fundamental parallelogram $P$ with vertices $0, 1, \omega, \omega+1\in {\mathbb C}$. Then the section is given by the map $\omega\to 0\in P$.) Then $M=E\_{\Gamma}\setminus \sigma$ is isomorphic to the quotient of the Teichmuller space $T(S)$ of $S$ by a finite-index torsion-free subgroup $Mod^o\_S$ of the mapping class group $Mod\_S$ of $S$. Now, this is a general fact (Deligne-Mumford, et al) that $T(S)/Mod^o\_S$ is quasi-projective (for any Riemann surface of finite type). It is not hard to see that DM compactification of $T(S)/Mod^o\_S$ is our case will add (among other things) the curve $\sigma$ back to $M$, thus, providing a projective compactification of $M$. 


In the special case you are interested in, it seems that quasi-projectivity of $E\_{\Gamma}$ was first proven by Kodaira (On compact analytic surfaces. II, III), at least, Shioda (On elliptic modular surfaces, 1972) attributes the result to him. 


Addendum: As an alternative to this argument, one can use Ron Livne's thesis "On certain covers of the universal elliptic curve". Livne proves that for every level $N\ge 5$ congruence subgroup $\Gamma(N)$ in the modular group $SL(2, {\mathbb Z})$, the universal elliptic curve over $E(N):={\mathbb H}^2/\Gamma(N)$ admits a degree $d\ge 2$ cyclic branched cover $E\_d(N)$, so that the latter admits a compactification $X\_d(N)$ (compatible with branched cover), so that $X\_d(N)$ is a general type projective surface. Thus, every $E\_\Gamma$ admits a finite regular cover which is biholomorphic to a finite cover over one of the $E\_d(N)$'s. Thus, $E\_\Gamma$ is quasi-projective. 


Livne also refers to Mumford's paper "Prym varieties. I." Contributions to analysis (a collection of papers dedicated to Lipman Bers), pp. 325–350. Academic Press, New York, 1974, for a direct proof of quasi-projectivity of $E(N)$'s. I do not have access to Mumford's paper, so I cannot say for sure. 


