We Ashkenazim have a minhag that the mothers of the Chasan and Kallah break an earthenware plate after the Tna'im is read. (I am not talking about the glass under the Chupah. That minhag stems from Brachos 31b, as Tosfos says there.
רב בריה דרבינא עבד הלולא לבריה חזנהו לרבנן דהוו קבדחי טובא אייתי כסא דמוקרא בת ארבע מאה זוזי ותבר קמייהו ואעציבו רב אשי עבד הלולא לבריה חזנהו לרבנן דהוו קא בדחי טובא אייתי כסא דזוגיתא חיורתא ותבר קמייהו ואעציבו אמרו ליה רבנן לרב המנונא זוטי בהלולא דמר בריה דרבינא לישרי לן מר אמר להו ווי לן דמיתנן ווי לן דמיתנןI'm not talking about that. I'm talking about breaking the plate at the Te'na'im.)
All our minhagim are holy and meaningful, but among those holy and meaningful minhagim, this one does not stand on the highest rung. But it's worth bearing in mind that, as I've said before, the meaning of our minhagim is fluid and dynamic; they ebb and flow. What a minhag means to one generation, to one group, might be very different than what it means to another. See, e.g., our discussion of the Kittel here, where we showed that wearing the kittel could symbolize diametrically opposed ideas, and that with time, one idea has become dominant, and our discussion of masks on Purim here. The symbol's meaning is what you understand it to be, and even minor minhagim can come to assume greater significance. The same is true regarding the breaking of the plate. Here's a list of the various interpretations that have attributed to it by our mefarshim. I'm listing all the time honored classics; I'M NOT IMPLYING ANY PARITY HERE! Some are stranger than others, several are similar but have differences in tone. Pick the one you like. They're all kosher.
The first written mention of this minhag is in the Sefer Ma'adanei Yomtov, written around 1600 by Reb Yomtov Lippman Heller, the author of the Tosfos Yomtov. He was a talmid of the Maharal. And he is the Ketzos' grandfather.
1. To temper the celebration Zeicher Le'Mikdash. Ma'adanei Yomtov (cited by Eliahu Rabba, which is cited by Pri Megadim in OC 560 Mishbetzos SK 7, and cited by Mishna Berura there SK 9, but I found it here) says the purpose is to shock the onlookers, in order to temper excessive joy that is inconsistent with mourning for the Churban Beis Hamikdash.
2. To show that the Te'na'im is irreversible, and whoever breaks it can never be made whole. The Pri Megadim brings the Maharit, which he understands to mean that although we break glass under the Chupa, we should break earthenware at the Tna'im, because glass can be melted and remade, but earthenware, once broken, can never be repaired. This is also said in the name of the Gaon in the She'iltos on the Ma'aseh Rav in numbers 133 and 134, here.
3. As a re-enactment of Mattan Torah, the breaking of the Luchos, and our ultimate redemption. The Pri Megadim himself says the following: Mattan Torah was Kidushin, and it ought to have been followed by Nesu'in. The sin of the Egel, followed by the breaking of the luchos, and ultimately the destruction of the Beis Hamikdash, left us almost bereft of the Hashra'as Hashechina that should have been ours as the beloved of Hakadosh Baruch Hu. But the day will come that Hashem will betroth us again, and that kidushin will be followed by a nisu'in and a permanent union. So we break earthenware at the Tna'im, to symbolize the breaking of the luchos and the impermanence of Klal Yisrael's first kiddushin, but we break glass at the chupa, because glass can always be repaired; in a sense, it can never be permanently or irreparably broken.
4. To show that breaking a tna'im is worse than getting divorced. The Gaon is quoted here as having said that it is worse to break a Tna'im than to get married and divorced. Or, that it is better to get married even if you know you're going to get divorced than to break a tna'im. So we use irreparable earthenware at the Tna'im, and glass at the Chuppah.
5. To show that the only one way to break a tna'im: by dying. The Baal Shem Tov is quoted in Taamei Haminhagim (page 411) as having explained this Maharit as meaning that a tna'im cannot be broken for any reason at all, but a marriage can be dissolved via a get.
6. To remind the chasan that even if his wife turns out to be a shrew, he should be grateful, because suffering through a miserable marriage will earn him a ticket straight to Olam Haba. (Hopefully, without her.) The Ta'amei Haminhagim also brings from the Likutei Maharan the following reason: you break an earthenware vessel at the Tna'im to remind the Chasan that there is a Gehinnom, and that he better not be mindlessly driven by his bodily desires. Also, he says, even if (Chas veshalom!) the answer to the question is "motzei," and it turns out that his wife is no good, he should still not "traitorously" divorce her, because his lifetime of suffering will save him from Gehinnom.
7. To prevent excessive frivolity so that we don't forget our Yiras Shamayim. See above from Brachos 31, Rav Hamnuna Zuti, who, when asked to sing at a wedding, sang "Woe to us, we all will die....." and the rule of Rav Yochanan/Rav Shimon bar Yochai,
א"ר יוחנן משום רשב"י אסור לאדם שימלא שחוק פיו בעולם הזהthat one may not "fill his mouth with joy in this world." Some learn that this 'reigning in of levity' is so that we should remember the Churban, that we should elevate Yerushalayim over our joys, as indicated in the Ramban in Toras Ha'adam. This approach is seen in the Tosfos Yomtov's pshat, #1 on this list, and that's why this halacha is discussed in Hilchos Tisha Ba'av. HOWEVER: Rabbeinu Yonah in Brachos there says that this rule applied even when the Beis Hamikdash stands, because immoderate levity is incompatible with Yir'as Shamayim. The same reasoning would apply to our discussion-- that the breaking of the plate at the Tna'im is more in line with Rav Hamnuna Zuti's dirge than it is with Zeicher Le'Mikdash.
8. Why do the mothers do this? Well, it seems that in some places, this was not done by the mothers. In this drawing from 1724, a man is doing it (the fellow bottom center with the jug raised above his head).
And, here's the bonus video. It's from a Sardinian wedding, and they're not Jews. There were Jews in Sardinia from before the destruction of the second Beis Hamikdash until the expulsion that accompanied the Inquisition in the late fifteenth century. If you want to believe they got the minhag from us, go ahead and believe it. I think people, Jews and Gentiles, do it because it's just fun to break stuff.
Now, you are all experts on this minhag. I hope this has cheered you up.