The Taguchi method is simply what in statistics is call a fractional factorial design. It enables estimations of the main effects and in some cases some of the interaction effects by making the assumption that higher order interactions are not present (effect=0). The 9 refers to the number of runs (experimental units) and the orthogonal array refers to the fact that the design is balanced so that one can get useful estimates of the main effects. It is an efficient design in that it doesn't require as many experimental units as a full factorial design (all possible combinations of factors and levels) but comes with the risk that the estimates may be biased if there is in fact an interaction effect. For a 3 by 3 (3 factors each with 3 levels) simply assign to each of the nine experimental units the level as indicated in the table to each of the 3 factors.

for three 3-level factors, you will have 27 combinations in factorial design. but you can also use part of them, fractional factorial design L9. you may refer to (1) (PDF) Robust Parameter Design of Shielded Metal Arc Welding (SMAW) for Optimum Tensile Strength (researchgate.net), how to use L9 in robust design.

... look for the advantages of RSM over taguchi approach