While the Sun and Earth attract each other, they cannot fall into each other because of angular momentum conservation. In a central field (where the force is acts in the direction of the distance vector and depends on distance only), the specific angular momentum vector $\boldsymbol{L}=\boldsymbol{r}\times\boldsymbol{v}$ is conserved ($\boldsymbol{r}$ is position and $\boldsymbol{v}$ the velocity). In particular $L=|\boldsymbol{L}|=rv_t$ (with $v_t$ the tangential velocity: the component of velocity perpendicular to the direction Earth-Sun). The specific orbital energy $E=\tfrac{1}{2}\boldsymbol{v}-GM/r$ is also conserved and must be negative for a bound orbit (such as Earth's). Combining these two we have $$ E = \tfrac{1}{2}v_r^2 + \frac{L^2}{2r^2} - \frac{GM}{r} $$ with $v_r$ the radial component of velocity. Since $v_r^2\ge0$, but $E<0$, not all radii are reachable. In particular there are two radii at which $v_r=0$: the apo- and peri-apse of the orbit.
In fact, for the Earth, the peri- and apo- apse are quite similar, and the orbit is nearly circular. The gravitational pull of the Sun is very nearly balanced by the centrifugal force due to the rotating orbit.
In GR, the picture doesn't really change much. Non-Newtonian (GR) effects are tiny in the Solar system and completely unimportant for Earth's orbit.