Let $N$ be a positive integer, $p$ be a prime not dividing $N$. It is a theorem of Igusa that the modular curve $X_1(N)$ has good reduction at $p$, call the reduction $\tilde{X_1}(N)$. If I'm understanding things correctly, this reduction should itself be representing some moduli problem (I'm already a little confused about how this works, but I don't think it matters so much for my main question). So points of $X_1(N)$ should correspond to elliptic curves $E$ with level structure, and the reduction map should send such a point to its reduction, again with some level structure.
My question then is, what happens to a point corresponding to an elliptic curve with bad reduction? It should be mapped to some point of $\tilde{X_1}(N)$, so it should correspond to some elliptic curve in characteristic $p$, but the reduction isn't an elliptic curve, so something must have gone wrong.
The closed curve $X_1(N)$ parameterizes not just elliptic curves with level structure (which correspond to the non-cuspidal points, the open subset of which is usually denoted $Y_1(N)$), but also degenerate elliptic curves (roughly speaking, nodal curves) with level structure (these are classified by the cusps).
The points on $Y_1(N)$ with bad reduction reduce to cusps on $\widetilde{X}_1(N)$.
A simple model to think about is $\mathbb A^1$ inside $\mathbb P^1$. A point in $\mathbb A^1$ over $\mathbb Q_p$ will have reduction lying in $\widetilde{\mathbb A}^1$ (here I'm using your notation; in this case I just mean the affine line over $\mathbb F_p$) if and only if it lies in $\mathbb Z_p$; elements of $\mathbb Q_p$ with non-trivial denominators have reduction equal to the point at infinity in $\widetilde{\mathbb P^1}$.
Another way to think about this is that $0$ is not the only element of $\mathbb Q_p$ whose reduction is zero mod $p$; any element of $\mathbb Q_p$ of positive valuation has reduction equal to zero mod $p$. Similarly, $\infty$ is not the only point of $\mathbb P^1(\mathbb Q_p)$ whose reduction is the point $\infty$ in $\mathbb P^1(\mathbb F_p)$; any element of $\mathbb Q_p$ with negative valuation has reduction equal to $\infty$.