There was a presentation of the idea at SUSY2013 by Nima Arkani-Hamed which is available on video at http://susy2013.ictp.it/video/05_Friday/2013_08_30_Arkani-Hamed_4-3.html
The amplituhedron is a buzzword for a description of a way to solve maximally supersymmetric (i.e. N=4) Yang-Mills theory in 4 dimensions. Ordinary Yang-Mills theory is a generalization of quantum gauge field theories which include electrodynamics and quantum chromodynamics. The supersymmetric extensions have not been found in nature so far.
The usual way to calculate scattering amplitudes in quantum field theory is by adding together the effects of many Feynman diagrams, but the number and complexity of diagrams increases rapidly as the number of loops increases and if the coupling is strong the sum converges slowly making it difficult to do accurate calculations.
The new solution for Super Yang-Mills uses the observation that the theory has a superconformal invariance in space-time and another dual superconformal invariance in momentum space. This constrains the form that the scattering amplitudes can take since they must be a representation of these symmetries. There are further constraints imposed by requirements of locality and unitarity and all these constraints together are sufficient to construct the scattering amplitudes in the planar limit without doing the sum over Feynman diagrams. The mathematical tools needed are twistors and grassmanians. The answer for each scattering amplitude takes the form of a volume of a high dimensional polytope defined by the positivity of grassmanians, hence the name amplituhedron.
The first thing to say about this is that so far it is only applicable to the planar limit of one specific quantum field theory and it is not one encountered in nature. It is therefore very premature to say that this makes conventional quantum field theory obsolete. Some parts of the theory can be generalised to more physical models such as QCD but only for the tree diagrams and the planar limit. There is some hope that the ideas can be broadened beyond the planar limit but that may be a long way off.
On its own the theory is very interesting but of limited use. The real excitement is in the idea that it extends in some way to theories which could be physical. Some progress has been made towards applying it in maximal supergravity theories, i.e. N=8 sugra in four dimensions. This is possible because of the observation that this theory is in some sense the square of the N=4 super Yang Mills theory. At one time (about 1980) N=8 SUGRA was considered a candidate theory of everything until it was noticed that its gauge group is too small and it does not have chiral fermions or room for symmetry breaking. Now it is just considered to be another toy model, albeit a very sophisticated one with gravity, gauge fields and matter in 4 dimensions. If it can be solved in terms of something like an amplituhedron it would be an even bigger breakthrough but it would still be unphysical.
The bigger hope then is that superstring theory also has enough supersymmetry for a similar idea to work. This would presumably require superstring theory to have the same dual superconformal symmetry as super Yang Mills, or some other even more elaborate infinite dimensional symmetry. Nothing like that is currently known.
Part of the story of the amplituhedron is the idea that space, time, locality and unitarity are emergent. This is exciting because people have always speculated that some of these things may be emergent in theories of quantum gravity. In my opinion it is too strong to call this emergence. Emergence of space-time implies that space and time are approximate and there are places such as a black hole singularity where they cease to be a smooth manifold. The amplituhedron does not give you this. I think it is more accurate to say that with the amplituhedron space-time is derived rather than emergent. It is possible that true emergence may be a feature in a wider generalisation of the theory especially if it can be applied to quantum gravity where emergence is expected to be a feature. Having space-time and unitarity as a derived concept may be a step towards emergence but it is not the same thing.
For what my opinion is worth I do think that this new way of looking at quantum field theories will turn out to something that generalises to something that is really part of nature. I have advocated the idea that string theory has very large symmetries in the form of necklace algebras so these ideas seem on the right track to me. However I do think that many more advances will be required to work up from super yang mills to sugra and then string theory. They will have to find a way to go beyond the planar limit, generalise to higher dimensions, include gravity and identify the relevant symmetries for string theory. Then there is just the little issue of relating the result to reality. It could be a long road.