I would like to ask the following question: Suppose an mdimensional manifold in an ndimensional euclidean space, choose some point on this manifold and take an ndimensional ball of radius R centred at this point. If the volume of the part of the manifold "enclosed" in this ball is V, what can we tell about the bounds for the different types of curvatures (scalar, mean, sectional curvatures) for the manifold inside this ball? Thanks very much in advance!

Not much if you do not restrict the relation between the dimensions $m$ and $n$. According to Nash' isometric embedding theorem (see his paper in Annals of Math. (2) 63, theorem 3), the isometric embedding can be obtained in an arbitrarily small ball in $\mathbb{R}^n$. 

