The short answer is nothing. While genuinely a Jewish institution, the word itself is not in Hebrew, but rather, in Greek.
Just like thousands of other loanwords, this word made its way into dozens of other languages. The word synagogue is never actually mentioned in the Hebrew Bible, but is frequently mentioned in the New Testament.
When we translate the word “synagogue” to Hebrew, the phrase we use to communicate this idea is בֵּית כְּנֶסֶת Beyt Kneset which literally means, “House of meeting.” If this sounds somewhat familiar it is most likely because you recognize the word “Knesset “which is the name used for Israeli parliament.
So what does Synagogue mean in Greek? Like its Hebrew counterpart, it simply means “meeting” or “gathering.”
In James 2:2 we read: “For if a man comes into your synagogue with a gold ring and dressed in fine clothes, and there also comes in a poor man in dirty clothes, and you pay special attention to the one who is wearing the fine clothes, and say, “You sit here in a good place,” and you say to the poor man, “You stand over there, or sit down by my footstool,” have you not made distinctions among yourselves, and become judges with evil motives?”
