This article explains the basic implementation in length:
The goal of this blog post is to show the fundamentals of A* through a really simple C# implementation.
It also points to better implementations, more suitable for production use:
As for ways to find better routes, there are plenty of C# examples around that are far better and richer than this one. CastorTiu has a really nice demo solution on CodeProject, A* algorithm implementation in C#, that animates the search algorithm and allows the user to tweak a few settings. […]
EpPathFinding.cs- A Fast Path Finding Algorithm (Jump Point Search) in C# (grid-based). It has a nice, clear GUI and allows a few settings to be tweaked.