How can I declare a two dimensional string array?

string[,] Tablero = new string[3,3];

You can also instantiate it in the same line with array initializer syntax as follows:

string[,] Tablero = new string[3, 3] {{"a","b","c"},
                                      {"d","e","f"}, 
                                      {"g","h","i"} };

You just declared a jagged array. Such kind of arrays can have different sizes for all dimensions. For example:

string[][] jaggedStrings =  {
new string[] {"x","y","z"},
new string[] {"x","y"},
new string[] {"x"}
};

In your case you need regular array. See answers above. More about jagged arrays

You probably want this:

string[,] Tablero = new string[3,3];

This will create you a matrix-like array where all rows have the same length.

The array in your sample is a so-called jagged array, i.e. an array of arrays where the elements can be of different size. A jagged array would have to be created in a different way:

string[][] Tablero = new string[3][];
for (int i = 0; i < Tablero.GetLength(0); i++)
{
    Tablero[i] = new string[3];
}

You can also use initializers to fill the array elements with data:

string[,] Tablero = new string[,]
{
    {"1.1", "1.2", "1.3"},
    {"2.1", "2.2", "2.3"},
    {"3.1", "3.2", "3.3"}
};

And in case of a jagged array:

string[][] Tablero = new string[][]
{
    new string[] {"1.1", "1.2"},
    new string[] {"2.1", "2.2", "2.3", "2.4"},
    new string[] {"3.1", "3.2", "3.3"}
};