Investigate the 8 fields arround every field in two dimensional array?

Yes, so you need to take account of that by checking for the row and column values being 0 or the height/width. If you're just adding up values, you may find it easiest to have a method which just returns the value in the array, or 0 if you're asking for a value out of range:

int GetValue(int[,] array, int x, int y)
{
    if (x < 0 || y < 0 ||
        x >= array.GetLength(0) || y >= array.GetLength(1))
    {
        return 0;
    }
    return array[x, y];
}

Then you can just use:

for (int x = 0; x < array.GetLength(0); x++)
{
    for (int y = 0; y < array.GetLength(0); y++)
    {
        int total = GetValue(array, x - 1, y - 1)
                  + GetValue(array, x, y - 1)
                  + GetValue(array, x + 1, y - 1)
                  + GetValue(array, x - 1, y)
                  + GetValue(array, x + 1, y)
                  + GetValue(array, x - 1, y + 1)
                  + GetValue(array, x, y + 1)
                  + GetValue(array, x + 1, y + 1);
        // Do something with the total
    }
}

Note that all the calls to GetLength are pretty inefficient here. There are alternatives which would be more efficient, but a bit more complicated.