The general form of the equation for the force of gravity on a particle of mass $m_2$ by another particle $m_1$ is
$$\vec{F}_{21}(\vec{r}) = \frac{G m_1 m_2}{|\vec{r}|^2} \hat{r}$$
where $\vec{r} \equiv \vec{r}_2 - \vec{r}_1$ is a vector from $m_1$ to $m_2$ and $G$ is the gravitational constant.
If we choose an arbitrary cartesian coordinate system, this vector becomes
$$\vec{r} = (x_2 - x_1)\hat{x} + (y_2 - y_1)\hat{y} + (z_2 - z_1)\hat{z}$$
and
\begin{align}
|\vec{r}|^2 &= \vec{r} \cdot \vec{r} \\
&= (x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2
\end{align}
From Newton's Laws, we can also express this force as an acceleration on $m_2$ due to $m_1$:
$$\vec{F}_{21}(\vec{r}) = m_2 \ddot{\vec{r}} = \frac{G m_1 m_2}{|\vec{r}|^2} \hat{r}$$
$$\ddot{\vec{r}}_{21} = \frac{G m_1}{|\vec{r}_{21}|^2} \hat{r}_{21}$$
For an ensemble of $N$ particles, we can apply the principle of superposition to compute the acceleration of a particle $m_i$
due to all other particles:
\begin{align}
\ddot{\vec{r}}_i &= G \sum_{j \neq i}^N \frac{m_j}{|\vec{r}_{ji}|^2} \hat{r}_{ji} \\
&= G \sum_{j \neq i}^N \frac{m_j}{|\vec{r}_{ji}|^2} \frac{\vec{r}_{ji}}{|\vec{r}_{ji}|} \\
&= G \sum_{j \neq i}^N \frac{m_j}{|\vec{r}_{ji}|^3} \vec{r}_{ji}
\end{align}
For practical purposes, it helps to break this down into its unit vector components (in cartesian coordinates):
\begin{align}
\ddot{\vec{x}}_i &= G \sum_{j \neq i}^N \frac{m_j}{|\vec{r}_{ji}|^3}(x_j - x_i)\hat{x} \\
\ddot{\vec{y}}_i &= G \sum_{j \neq i}^N \frac{m_j}{|\vec{r}_{ji}|^3}(y_j - y_i)\hat{y} \\
\ddot{\vec{z}}_i &= G \sum_{j \neq i}^N \frac{m_j}{|\vec{r}_{ji}|^3}(z_j - z_i)\hat{z}
\end{align}
To integrate these equations numerically, we need to further break down this set of *three* 2nd-order ODEs
into a set of *six* 1st-order ODEs:
\begin{align}
\dot{x}_i &= u_i \\
\dot{y}_i &= v_i \\
\dot{z}_i &= w_i \\
\dot{u}_i &= G \sum_{j \neq i}^N \frac{m_j}{|\vec{r}_{ji}|^3}(x_j - x_i) \\
\dot{v}_i &= G \sum_{j \neq i}^N \frac{m_j}{|\vec{r}_{ji}|^3}(y_j - y_i) \\
\dot{w}_i &= G \sum_{j \neq i}^N \frac{m_j}{|\vec{r}_{ji}|^3}(z_j - z_i)
\end{align}
where
$|\vec{r}_{ji}|^3 = \big[ (x_j - x_i)^2 + (y_j - y_i)^2 + (z_j - z_i)^2 \big]^{3/2}$
We can now apply a numerical integrator (here I've chosen a fourth-order Runge-Kutta, or RK4) to see the time evolution
of this set of equations. Let's write all of our equations as a single *vector* at time step $n$ which we can
operate on: $\vec{y}_n \equiv \{ {x}_i, {y}_i, {z}_i, {u}_i, {v}_i, {w}_i : i \in N\}$, whose *derivative*
is $\vec{f}\prime(t_n, \vec{y}_n) = \{ \dot{x}_i, \dot{y}_i, \dot{z}_i, \dot{u}_i, \dot{v}_i, \dot{w}_i : i \in N\}$.
Then, given a time step interval $h$, our RK4 method will do the following algorithm:
$$\vec{y}_{n+1} = \vec{y}_n + \frac{1}{6} h (k_1 + 2 k_2 + 2 k_3 + k_4)$$
where
\begin{align}
k_1 &= \vec{f}\prime(t_n, \vec{y}_n) \\
k_2 &= \vec{f}\prime(t_n + h/2, \vec{y}_n + h k_1/2) \\
k_3 &= \vec{f}\prime(t_n + h/2, \vec{y}_n + h k_2/2) \\
k_4 &= \vec{f}\prime(t_n + h, \vec{y}_n + h k_3)
\end{align}
For the following animations, I implemented these equations in the Julia language
language, and then created the visualizations with Python Matplotlib and ffmpeg.

Under certain conditions in a three-body system, a dynamical phenomenon known as the Kozai Mechanism can cause a distant third body to dramatically affect the orbit of an otherwise stable inner binary. On a timescale much longer than the orbital period, the perturbing third body drives a periodic exchange between the inner system's eccentricity and inclination. This process may be an important factor in planet evolution and migration, and may be a key factor in explaining how hot Jupiters form.

The first animation shows a three-body simulation from the rest-frame comoving with the system's center-of-mass, with two massive objects in orbit around each other and a test particle orbiting the inner, more massive star. The axes show the spatial scale in AU and the timescale is shown at the top. The masses and orbital parameters of all three bodies were selected to optimize viewability for the purpose of the demonstration. The second animation shows the same simulation, only transformed into the rest-frame of the more massive central star. Here, it becomes apparent that the orbit of the test mass is perturbed by the outermost star and we can see the periodic changes in the inclination and eccentricity of its orbit.

This animation depicts the TRAPPIST-1 system, a distant solar system with at least seven terrestrial planets orbiting a cool M-dwarf sun.

Here are a few more N-body simulations.

The first animation shows a hypothetical exoplanet system with two planets in stable orbits around a central binary star (e.g., the circumbinary planet Kepler-16b.) The second animation shows a hypothetical planet on a chaotic trajectory through a binary star system. The planet is ultimately ejected from the system.