I know about curvature by this notation $$\tau=\frac{dt}{ds}$$ the change of tangent vector with respect to arc length $s$ .
I also know about Ricci scalar curvature is $$g^{ij}R_{ij}=R$$
I know the formulas. But i want to really know about their differences and some helpful geometrical interpretations.
Your formula $\tau = \frac{dt}{ds}$ requires a correction, namely $t$ is not just any old tangent vector but is instead the unit length tangent vector. Also, that notion of curvature, which is known as "geodesic curvature", applies only to curves (1-dimensional objects) in space, the one parameter being $s$.
On the other hand Ricci curvature applies only to 2-or-higher dimensional objects in space.
So, you're not going to find much in the way of a direct comparison between those two types of curvature.
Still, though, there are some indirect comparisons in some limited situations. One particularly close connection occurs for a 2-dimensional surface $S$ in 3-dimensional space. The Ricci curvature at a point $P \in S$ is equal to the Gaussian curvature (because in 2 dimensions there's nothing to contract in the contraction formula that you give). And the Gaussian curvature is equal to the product of two different geodesic curvatures, namely the so-called "principle curvatures" which are the maximum and minimum values of $\tau$ for curves passing through $P$.
