For any $n\geq 1$, one can define a functor $\mathcal{M}_1(n)\colon \mathrm{Schemes}/\mathbb{Z}[\frac1n] \to \mathrm{Groupoids}$, sending a scheme to the groupoid of elliptic curves over it with a chosen point of exact order $n$; the morphisms in the groupoid are the isomorphisms of elliptic curves that respect the chosen point. If $n\geq 4$, the groupoid is equivalent to a set (i.e. there can be no non-trivial automorphism of an elliptic curve that preserves a point of order $n\geq 4$) and the functor is actually represented by a scheme. This scheme might be called $Y_1(n)$. If you define it like this, the universal elliptic curve over $Y_1(n)$ is simply classified by the identity. (For $n<4$, the scheme $Y_1(n)$ is the coarse moduli space of the stack $\mathcal{M}_1(n)$ and the story becomes more complicated.)

But maybe $Y_1(n)$ has another definition for you, e.g. as the quotient of the upper half plane $\mathbb{H}$ by $\Gamma_1(n)$. It is a non-trivial theorem to identify this with the analytification of the $Y_1(n)$ as constructed above. But you can construct the universal elliptic curve in this case also easily in a direct way: Just take the quotient of $\mathbb{H} \times \mathbb{C}$ by the suitable semidirect product of $\Gamma_1(n)$ and $\mathbb{Z}^2$.

Probably, Katz's $p$-adic properties of modular schemes and modular forms is not the worst place to read about this. From the purely complex point of view you also have Hain's Lectures on Moduli Spaces of Elliptic Curves.

Regarding explicit Weierstraß equations: in general, this is bound to be difficult as even explicit equations for the $Y_1(n)$ are hard to get. An older article which deals in particular with such equations is *Torsion groups of elliptic curves with integral j-invariant over quadratic fields* by Hans H. Müller, Harald Ströher and Horst G. Zimmer. Regardless of the precise equations for $Y_1(n)$ ones knows however that the universal elliptic curve over it always has a Weierstraß equation of the form
$$y^2 + a_1xy + a_3y = x^3 +a_2x^2,$$
for holomorphic modular forms $a_1, a_2$ and $a_3$ of level $n$ over $\mathbb{Z}[\frac1n]$. This might have been known before, but we give a proof in an article with V. Ozornova: Rings of modular forms and a splitting for $TMF_0(7)$. There we also work out the case $n=7$ in detail.