NEAT is NeuroEvolution of Augmenting Topologies; an evolutionary algorithm devised by Kenneth O. Stanley.
SharpNEAT is a complete implementation of NEAT written in C# and targeting .NET (on both MS Windows and Mono/Linux).
From the SharpNEAT FAQ...
In a nutshell, SharpNEAT provides an implementation of an Evolutionary Algorithm (EA) with the specific goal of evolving neural networks. The EA uses the evolutionary mechanisms of mutation, recombination and selection to search for neural networks with behaviour that satisfies some formally defined problem. Example problems might be how to control the limbs of a simple biped or quadruped to make it walk, how to control a rocket to maintain vertical flight, or finding a network that implements some desired digital logic (such as a multiplexer).
A notable point is that NEAT and SharpNEAT search both neural network structure (network nodes and connectivity) and connection weights (inter-node connection strength). This is distinct from algorithms such as back-propogation that generally attempt to discover good connection weights for a given structure.
SharpNEAT is a kit of parts that facilitate research into evolutionary computation and specifically evolution of neural networks. The code as released provides a number of example problem domains that demonstrate how the various parts can be wired together to produce a complete working EA. Alternatively researchers can swap in alternative parts, such as a new/alternative genetic coding or even a new evolutionary algorithm. The provision for such rewiring was a major design goal of SharpNEAT and is facilitated by abstractions made in SharpNEAT's architecture around key concepts such as genome (genetic representation and coding) and evolutionary algorithm (mutations, recombination, selection strategy).
Motivation for the development of SharpNEAT mainly came from a broader interest in biological evolution, and more specifically curiosity on what the limits of neuro-evolution are in terms of the level of problem complexity it can produce satisfactory solutions for.