This is a high-level introduction to the topic. It assumes familiarity with calculus and special relativity. It’s also very incomplete, I just thought the equations would look cool in the overall aesthetic of this website lol. Finally, it’s also incoherent because general relativity is hard and it would take a lot of effort to put it in a reasonable order, which I will do at some point!
Newtonian gravitation states that the relationship between mass and the gravitational field it generates is
$$\nabla^2\Phi = 4\pi G\rho$$
The force is, as usual, the negative gradient. For two point masses, this comes out as
$$\begin{aligned} F_G&=-\nabla\Phi\&=-\frac{GMm}{4\pi r^2} \end{aligned}$$
We can calculate the field for an arbitrary mass distribution $\rho(\vec{r})$:
$$\Phi(\vec{r},t)=-G\int{\frac{\rho(\vec{r}’,t)}{\left\lvert\vec{r}-\vec{r}’\right\rvert}d^3r’}$$
However, this clashes with special relativity! If the $t$ on both sides were the same, changing the position of a particle somewhere would instantaneously change the field everywhere, violating the speed of causality.
However, we’ve dealt with something like this before! The electromagnetic wave equation is
$$\square \phi=-\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2}+\nabla^2\phi=-\frac{\rho}{\varepsilon_0}$$
With solution
$$\phi(\vec{r},t)=\frac{1}{\varepsilon_0}\int{\frac{\rho(\vec{r’}, t-R/c)}{R}d^3r’}, \quad R=\left\lvert\vec{r}-\vec{r}’\right\rvert$$
The times are no longer the same, and the signal travels at the speed of light!
Surely we know how to fix Newtonian gravity now: we just replace $\nabla^2$ with $\square$ to get
$$\square\Phi=4\pi\rho G$$
However, we get two problems:
We must look for a new theory.
The weak equivalence principle states that it is impossible to detect a gravitational field in a local inertial frame (or equivalently, the ratio of gravitational mass to inertial mass is a constant).
The strong form states that all physics is independent of whether you are accelerating or in a local gravitational field.
This has an interesting quantum point, where phase evolves differently if you’re in a field, but this is only measurable by using a comparison point outside the field, thus not breaking the principle!
Newton’s first law can be generalised. In local inertial coordinates, the equation of motion is simply: no acceleration.
$$\frac{d^2\xi^\alpha}{d\tau^2}=0$$
A generalisation of the special relativistic spacetime interval gives us
$$-c^2d\tau^2=\eta_{\alpha\beta}d\xi^\alpha d\xi^\beta$$
If we switch to general coordinates $x^\mu$ using
$$\frac{d\xi^\alpha}{d\tau}=\frac{d\xi^\alpha}{dx^\mu}\frac{dx^\mu}{d\tau}$$
Then the equation of motion becomes
$$0=\frac{d^2\xi^\alpha}{d\tau^2}=\frac{d}{d\tau}\left(\frac{d\xi^\alpha}{dx^\mu}\frac{dx^\mu}{d\tau}\right)=\frac{dx^\mu}{d\tau}\frac{d}{d\tau}\frac{d\xi^\alpha}{dx^\mu}+\frac{d\xi^\alpha}{dx^\mu}\frac{d^2 x^\mu}{d\tau^2}$$
After some wrangling, this simplifies to
$$0=\frac{d^2 x^\lambda}{d\tau^2}+\left(\frac{\partial x^\lambda}{\partial\xi^\alpha}\frac{\partial^2\xi^\alpha}{\partial x^\lambda \partial x^\nu}\right)\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}$$
If we then define the affine connection, or the Christoffel symbol (the latter is a specialisation of the former to mainfolds with a metric),
$$\boxed{\Gamma^\lambda_{\mu\nu}=\frac{\partial x^\lambda}{\partial \xi^\alpha}\frac{\partial^2 \xi^\alpha}{\partial x^\mu\partial x^\nu}}$$
This becomes
$$\boxed{\ddot{x}^\lambda+\Gamma^\lambda_{\mu\nu}\dot{x}^\mu \dot{x}^\nu=0}$$
This is the equation of motion in a curved spacetime!