What's a Shabbat Kallah? It's a celebration for the bride on the Shabbos afternoon before her wedding day.
In some ways, it's like a frum bachelorette party. (Note that there may also be a "normal" bachelorette party with the bride and her girlfriends!)
I went to my first Shabbat Kallah last month, and it was a very lovely experience! However, classical feminists will probably balk at the experience. There is a terribly strong emphasis on pleasing a husband and having tons of babies. I admit, I was a bit put-off at first, and I am certainly not a classical feminist. However, the female solidarity and togetherness in that room was incredible!
There was food and drink out, and several women brought their young children. A rebbetzin served as the "MC" of the festivities, and I learned later that she is the kallah teacher (kallah teachers are another post, but to give you an idea, they're the women who normally teach an engaged woman about the laws of family purity). She hosted different "games" and then there was dancing. I left at that point, but the main festivities appeared to be over.
Give the kallah a bracha (blessing): The rebbetzin passed around a bag of random items. We each closed our eyes and selected something from the bag. The "game" is making up a bracha for the kallah based upon the item you chose! For instance, I drew a soup ladle, and my friend pulled out a toy camera. This was a really cool idea, and I was amazed by the creativity in the room! Thankfully, my friend is a bracha master, and she helped several of us formulate a bracha! As you might imagine (this is a room of orthodox women after all), a lot of the brachas came back to "have lots of babies and have them soon." That's very much a cultural thing in the orthodox Jewish world. It's still present in the liberal Jewish world, though less in-your-face.
Give the kallah marriage advice: The rebbetzin then had each of us choose a random book about marriage. The kallah chose a number, and we each turned to that page and came up with some marriage advice for the bride. I liked this "game" a lot, but some of the advice given would set a classical feminist foaming at the mouth. In retrospect, it really wasn't that bad, but it felt pretty sexist at the time. I later realized that I had expected the advice to be sexist after all the child-based brachot, so that's how I interpreted the advice at the time. However, I could generally notice a difference between the words of women from different generations, though that isn't a rule. The Jewish world, even the orthodox one, is a-changin'.
In summary, if you're invited to a Shabbat Kallah, GO. You don't need to know anything special. You don't even need to know the bride! I didn't. Just sit back and enjoy the festivities, and if you have questions, ask your neighbor. There's such a joy in the room that everyone's patience is at a high-point. (And of interest to converts: the attention is so focused on the bride that it's unlikely that anyone will think to question you!)