All objects follow paths in curved spacetime, and these paths are called geodesics.
An object falling in the earth's surface follows a geodesic that depends on its velocity and not its mass (light and other massless objects, where $v=c$ always, follow geodesics).
Suppose you define a Euclidean coordinate system on earth with you at the origin, and suppose you throw an object up into the air at $t = 0$.
The position of the object in spacetime is given by functions of an affine parameter, so that the path the object takes makes it appear like it is accelerating to earth, giving rise to the idea that gravity is a force. In reality, the motion of the object in the coordinate system is described by the geodesic equation
$$\frac{d^2x^{\alpha}}{dt^2}+\Gamma^\alpha_{\mu\nu}\frac{dx^{\mu}}{dt}\frac{dx^{\nu}}{dt}=0$$
where $x^\mu \,, \mu\in[0,1,2,3]$ is the position of the object in the coordinate system and $t$ is the proper time or the time measured by a clock following the object.
The first term on the LHS is the acceleration of the object in our coordinate system. The second term describes the effects of gravity where $\Gamma^\alpha_{\mu\nu}$ are called the Christoffel, or connection, symbols, and the other two are velocity. These symbols encode the effects of the curvature of spacetime. It shows us that the curvature of spacetime changes the acceleration of the object, based on its velocity through spacetime.
If there is no spacetime curvature, then all of the Christoffel symbols vanish $$\Gamma^\alpha_{\mu\nu}=0\leftrightarrow \frac{d^2x^{\alpha}}{dt^2}=0$$ and there is no acceleration (Christoffel symbols may not completely vanish in non-Euclidean coordinates, but this changes none of the physics). The important point is that the acceleration of the object is completely determined by the curvature of spacetime$^1$.
You may have heard this many times in general relativity, and that is matter curves spacetime, and the effect of the curvature of spacetime is to cause gravitational force. But it isn't a force that acts on the object, but instead it’s that the object follows a geodesic in spacetime.
$^1$ And the Einstein field equations tell us that the amount of spacetime curvature depends on the matter and energy content in the region of the spacetime, or $$G_{\mu \nu }+\Lambda g_{\mu\nu}= 8\pi G T_{\mu \nu }$$ where the LHS represents the curvature of spacetime and the RHS represents the stress–mass/energy–momentum content of the spacetime.
