A simplex (in $\mathbb{R}^{n}$) is a special case of a convex polytope, namely one with exactly $n+1$ extremal points (the convex hull of $n+1$ points in general position - that is to say they don't lie in a $(n-1)$-dimensional affine subspace). An example of a convex polytope which isn't a simplex is given by any one of the platonic solids except the tetrahedron.
Yes, there are more general notions of simplices. In a geodesic metric space, for example, it makes sense to speak of convex hulls. In hyperbolic $n$-space or on the $n$-dimensional sphere one can then define a hyperbolic or spherical simplex as the convex hull $(n+1)$ points that don't lie in a totally geodesic submanifold (and that are not too far apart from each other in the case of a sphere).