Why is the Synagogue important to Jews in the 21st century?
The Synagogue today is as relevant as it was throughout the ages. It is the place that Jews pray to G-d and connect to Him. Our sages tell us that when the holy temple was destroyed, the divine presence went from the holy temple, to the Synagogue, (obviously in a smaller form), and the Synagogue is even called a “mikdash me’at” a small temple.
The Talmud says that nowadays that the temple was destroyed, and we can no longer service G-d in the temple and bring the sacrifices, however the substitute for it is the prayers in the synagogue. Therefore when Jews come together to pray they want to do it specifically in a Synagogue, because it is a holy place.
On a lower level, the Synagogue is the center for Jewish activity, the place that Jews come together, and it helps them identify with Judaism, and makes them feel connected to G-d and to the Jews.
Tractate Megillah 29a.