General relativity changes the language of Newtonian mechanics at a fundamental level, such that it is no longer completely straightforward to describe what a force is.
To see this, consider that Newton's second law involves the acceleration
$$
m \frac{d^2 x^i}{dt^2} = F^i
$$
where $x^i=(x, y, z)$ is a position vector in three-dimensional space, and $F^i$ is the force vector. In Newtonian mechanics, there is an absolute time that everyone agrees on. However, even in special relativity, this is no longer true, because of the relativity of simultaneity.
Now, in special relativity, there is a more or less straightforward solution. So long as we stick to inertial observers, we can define a quantity called the "four-acceleration", $A^\mu$, where now $\mu=(t,x,y,z)$ is a label for a four-dimensional vector. The four-acceleration reduces to the usual notion of acceleration in the small-velocity limit, and otherwise is related to the change of velocity of some object, according to an inertial observer. We can set the rest mass times four-acceleration equal to a generalization of the force called the four-force, and end up with a version of Newton's second law.
But now we need to take another step, to general relativity, and now we can no longer assume there are inertial observers in the same way as in special relativity. There are locally inertial observers -- in other words, over small regions of spacetime, you can pretend you are working with special relativity, just like you can pretend geometry is Euclidean so long as you only work on a small patch of the Earth where you can ignore the curvataure. But we want a formulation that will work for any observer, and relate different inertial observers.
The generalization of Newton's second law that includes the effect of non-inertial coordinates is the geodesic equation
$$
\frac{d^2x^\mu}{d\tau^2} + \Gamma^{\mu}_{\nu\lambda} \frac{dx^\nu}{d\tau}
\frac{dx^\lambda}{d\tau} = 0
$$
where $\Gamma^\mu_{\nu\lambda}$ are the Christoffel symbols, $x^\mu=(t,x,y,z)$ is a set of coordinates in four dimensional spacetime, and $\tau$ is proper time (a parameterization of a curve, such that $g_{\mu\nu} \frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}=-1$, where $g_{\mu\nu}$ is the metric tensor and I'm using a mostly plus sign convention). This equation can be rearranged to be
$$
\frac{d^2x^\mu}{d\tau^2} = - \Gamma^{\mu}_{\nu\lambda} \frac{dx^\nu}{d\tau}
\frac{dx^\lambda}{d\tau}
$$
which is analogous to the force law we started with, and in an appropriate limit, the "$\Gamma$" term does reduce to the Newtonian force in the way you expect. So you could say that general relativity does contain an idea of a gravitational force on particles that is analogous to the Newtonian one.
However, from the more abstract perspective we need to even be able to talk about any of these quantities at all, the "$\Gamma$" term is part of what is called a "covariant derivative" and arises due to the fact that we need to consider non-inertial coordinates. Even without gravity, you can choose coordinates where this term is non-zero (essentially amounting to having fictitious forces in non-inertial frames). But gravity forces you to have this term, due to the presence of spacetime curvature. Therefore, it's also valid to say that spacetime curvature is the real reflection of gravity, and "force" is just an old-fashioned concept that is awkward to formulate in the new and more powerful language we need to use.
If GR were the end to the story, we would probably just say that gravity is spacetime curvature, and "force" is a word that applied to an old formulation of physics, and can be given meaning in the new and better formalism, but only in an ugly way. However, since we ultimately expect to be able to unify gravity with the other forces of nature, it seems like there should be some unified picture where we can talk about the forces in the standard model and gravity in the same language. Whether that requires us to reformulate the standard model (and quantum mechanics) in a more geometric language, or reformulate general relativity in a more "quantum interaction" based language,
or something else, remains to be seen.
