This is the special orthogonal group over the field $\mathbb{Z}/p \mathbb{Z}$. Or, rather, one of the "special orthogonal groups". For any invertible symmetric matrix $Q$, one can consider the group of matrices $U$ in $\text{SL}_n$ such that $U Q U^T = Q$, which forms a subgroup $\text{SO}(Q)$. If one allows matrices in $\text{GL}_n$ instead, this is $O(Q)$.

I'll limit myself to $p$ odd throughout.

**"Split" versus "non-split"** If $n=2m+1$ is odd, then any two nondegenerate quadratic forms define the same orthogonal group.

If $n=2m$ is even, then there are two classes of quadratic forms, called the "split" and the "non-split" class. The case you have described is split if $p^{m} \equiv 1 \bmod 4$ and is non-split if $p^{m} \equiv 3 \bmod 4$. I'll denote the split and non-split cases by $SO_{2m}^+(p)$ and $SO_{2m}^-(p)$.

**Order of the group** The order of $SO_{2m+1}$ is
$$p^{m^2} \prod_{i=1}^m (p^{2i}-1).$$
The orders of $SO_{2m}^+(p)$ and $SO_{2m}^-(p)$ are
$$p^{m(m-1)} (p^m-1) \prod_{i=1}^{m-1} (p^{2i}-1) \ \text{and} \ p^{m(m-1)} (p^m+1) \prod_{i=1}^{m-1} (p^{2i}-1)$$
respectively. Note, in particular, that they can't be isomorphic since they have different sizes.

**Generators** The orthogonal group $O_n$ is generated by reflections; for a vector $x$ with $x \cdot x \neq 0$, the reflection over $x$ is the linear map $r_x(v) := v - \tfrac{x \cdot v}{x \cdot x} x$. Reflections have determinant $-1$, so the special orthogonal group $SO_n$ is generated by pairs of reflections; the product $r_x r_y$ fixes the $(n-2)$-plane $x^{\perp} \cap y^{\perp}$, so we can also say that $SO_n$ is generated by rotations fixing planes of dimension $n-2$. A shorter list of generators can be found in Matrix Generators for the Orthogonal Groups, Rylands and Taylor , 1998.

**Structure** The orthogonal group $O_n$ has two characters to $\{ \pm 1 \}$: The determinant map, which sends every reflection to $-1$, and the spinor norm, which sends the reflection $r_x$ to $\left( \tfrac{x \cdot x}{p} \right)$ (this is the quadratic residue symbol). I'll write $K_{2m+1}$, $K_{2m}^+$ or $K_{2m}^-$ for the common kernel of these characters. So $K$ is an indexed $2$ subgroup of the corresponding $SO$.

For $n=2m+1$ odd, the group $K_n$ is the Chevalley group $B_m(p)$. The group $B_1(p)$ is isomorphic to $\text{PSL}_2(p)$, which is the alternating group $A_4$ if $p =3$ and is otherwise simple. I believe that $B_m(p)$ is simple for all $m \geq 2$ and $p \geq 3$, but I couldn't find a reference for this.

For $n$ even in the case you care about, where $Q = \text{Id}_n$, the group $K_n$ contains $- \text{Id}_n$, which generates the center. For the other quadratic form that you didn't use, the spinor norm of $- \text{Id}_n$ is $-1$, and the center of $K_{n}$ is trivial.

The quotients of $K_{2m}^+(p)$ and $K_{2m}^-(p)$ by their centers are the Chevalley group $D_m(p)$ and the twisted Chevalley group ${}^2 D_m(p^2)$ respectively. We have $D_2(p) \cong \text{PSL}_2(p) \times \text{PSL}_2(p)$. Again, $\text{PSL}_2(p)$ is simple for any $p \geq 5$, whereas $\text{PSL}_2(3) \cong A_4$. I believe that $D_m(p)$ and ${}^2 D_m(p^2)$ are simple for $m \geq 3$ (and $p$ odd), but again I couldn't find a reference

I'm using Wikipedia as my source for group theoretic notation.