API documentation for Gdal/Ogr with C#

This directory in the GDAL project tree has some C# examples, specifically this one that might suit your needs.

EDIT #1: The links below are now dead (left for posterity).
You can find the GDAL API documentation here
There is an GDAL API tutorial here
The OGR API documentation is here
The OSR API documentation is here
All the links you need for OGR/OSR can be reached from http://www.gdal.org/ogr/

EDIT 2: This link is still alive but C# info has quietly dissappeared
All the links you need for GDAL can be reached from https://gdal.org/

EDIT #3:
This page provides some C# links that still work (as of March 2020) but examples and documentation appears scant.