Using IComparer for sorting

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You need to implement the strongly type interface (MSDN).

public class CoordinatesBasedComparer : IComparer<Point>
{
    public int Compare(Point a, Point b)
    {
        if ((a.x == b.x) && (a.y == b.y))
            return 0;
        if ((a.x < b.x) || ((a.x == b.x) && (a.y < b.y)))
            return -1;

        return 1;
    }
}

BTW, I think you use too many braces, I believe they should be used only when they contribute to the compiler. This is my version:

if (a.x == b.x && a.y == b.y)
    return 0;
if (a.x < b.x || (a.x == b.x && a.y < b.y))
    return -1;

Just like I dislike people using return (0).

Note that if you target a .Net-3.5+ application you can use LINQ which is easier and even faster with sorting.

LINQ vesion can be something like:

var orderedList = Points.OrderBy(point => point.x)
                        .ThenBy(point => point.y)
                        .ToList();

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