What is the relationship between PyTorch and Torch?
There are two PyTorch repositories :

https://github.com/hughperkins/pytorch
https://github.com/pytorch/pytorch

The first clearly requires Torch and lua and is a wrapper, but the second doesn't make any reference to the Torch project except with its name.
How is it related to the Lua Torch?

Here a short comparison on pytorch and torch.
Torch:

A Tensor library like numpy, unlike numpy it has strong GPU support.
Lua is a wrapper for Torch (Yes! you need to have a good understanding of Lua), and for that you will need LuaRocks package manager.

PyTorch:

No need for the LuaRocks package manager, no need to write code in Lua. And because we are using Python, we can develop Deep Learning models with utmost flexibility. We can also exploit major Python packages likes scipy, numpy, matplotlib and Cython with PyTorch's own autograd.

There is a detailed discussion on this on pytorch forum. Adding to that both PyTorch and Torch use THNN. Torch provides lua wrappers to the THNN library while Pytorch provides Python wrappers for the same.
PyTorch's recurrent nets, weight sharing and memory usage with the flexibility of interfacing with C, and the current speed of Torch.
For more insights, have a look at this discussion session here.
