Why not define tensors under Galilean or Poincare transformations? I have seen vectors (and tensors, in general) defined under rotations,
$$V^i=R^i_{~j}V^j$$
and under Lorentz transformations,
$$V^{\prime\mu}=\Lambda^\mu_{~~\nu}V^\nu$$
where $R,\Lambda$ are the matrices representing rotations and Lorentz transformations.
My definition of tensors is old-fashioned i.e., in terms of how a bunch of numbers transforms under rotation or Lorentz transformations.
Why doesn't one talk about vectors (and tensors, in general) under the Galilean or Poincare transformations (or groups)? Wouldn't that be more general? Why doesn't one define tensors under translations, spatial, or spacetime? Please do not use more sophisticated definitions of tensors as it will be very painful for me to follow.
 A: 
Why doesn't one talk about vectors under the Galilean or Poincare transformations (or groups)? Wouldn't that be more general?

The tuple $(V^1,\ldots,V^n)$ is interpreted as the representation of some vector $V$ w.r.t. a basis. That being said, if we fix two bases, then the transformation of vector components is obviously linear (e.g. a rotation or a Lorentz transformation as you said). But Galilean transformations and Poincaré transformations are affine transformations.
The generalization you are looking for is the general linear group, the transformations between two arbitary bases. The point is that euclidean spaces and Minkowski space do not only have an affine structure, but also come with a bilinear form on the translation space. Each chart has an orthonormal basis associated to it and the transformation of vector components from one coordinate system to another is precisely the differential of the coordinate transformation (i.e. a Lorentz transformation in the case of Minkowski space).
