The periodic table is arranged in rows and columns in which the elements have similar properties. When Mendeleev created the table in the late 1800s, he did so because he had noticed patterns in the elements that were known at the time; there were repeating characteristics, as elements increased in atomic weight. When he arranged the elements then known, he was able to leave gaps in appropriate places. Even though those elements had not yet been discovered, he knew that there had to be one that fit into that slot--and he was right. Some were discovered soon after.
As far as predictive powers, elements in the same row (period) decrease in atomic radius as you move from left to right. Elements to the right of the diagonal line dividing the table separate elements generally tending to be metals (on the left) from non-metals (to the right). You can also predict which elements will combine with others, and in what ratios, because the number of electrons in the shells can be deduced.