The songs of Fiddler on the Roof are very distinctive, but it is also the primary means through which information is conveyed. At most times, there is at least background music, if not singing.
The choice of songs using notes from minor keys is very reminiscent of Jewish music. Some examples include Israel's national anthem, Hatikvah, which is one of the only national anthems in a minor key. Shalom Aleichem (which may look familiar-- Sholem Aleichem took his nom de plume from the common phrase of greeting!) is a common liturgical piece on Shabbat, sung after returning home from synagogue. There are many, many other examples of Jewish melodies which use a minor key, like Fiddler on the Roof does.
Below is a brief guide to the plot of the musical, in narrative form. It includes links to the songs to discuss more in depth some of the individual songs, their meaning, and the mechanisms employed. The links are in green, if you would prefer to skip the narrative.
⋯ACT 1⋯
The play opens with a musical number (Prologue/Tradition), which establishes the town of Anatevka, daily life, and Tevye's general character. We learn some key pieces of information: obviously, tradition is important to the insular Jewish community; there are some concerns about the bigger, non-Jewish village causing trouble but they're doing their best to maintain the peace.
Act 1 Scene 1 opens on the women of Tevye's household preparing for Shabbos. The girls warn their mother, Golde, that Yente the Matchmaker is coming, and Golde sends them out of the house. It is revealed that Yente has matched Tzeitel with Lazar Wolf, the Butcher. When the girls are allowed back inside, Tzeitel expresses her distress at the concept of being matched, and the girls have a song about matchmaking (Matchmaker). This begins to set the stage, so to speak, for the concept that perhaps Tevye's love of Tradition, expressed in the prologue, is no longer the most important thing for the next generation.
When Tevye arrives home, he is no longer in such a good mood, as he's had to pull his own cart-- his horse lost his shoe, and had to go to the blacksmith's. As work is not permitted on Shabbos, presumably Tevye will be without his horse for at least few days. He laments his life, and imagines a reality where he wasn't poor, questioning why he couldn't have a little more wealth (If I were a Rich Man). This is the first hint that Tevye's goal in life is mainly to elevate his daughters' social standing a little bit from his own. He is confronted by people who didn't receive their orders for Shabbos, and meets a stranger, Perchik. Perchik is a teacher, and a political radical, but in exchange for giving his daughter lessons, Tevye will feed him-- a beneficial agreement for them both.
While Tevye (and Perchik) goes inside to prepare for Shabbos, Golde tries to get him to talk to Lazar Wolf, who wishes to marry Tzeitel. Tevye misunderstands, and assumes Lazar Wolf wants to buy his milk cow. Golde doesn't dissuade him from this misconception. Motel the Tailor, who is Tzeitel's beloved, tries to talk to Tevye about marriage but struggles to find courage. Eventually, though, it is Shabbos, and they sing a prayer (Sabbath Prayer) while lighting the Shabbos candles.
After Shabbos, we see Tevye and Lazar Wolf meet to discuss Lazar Wolf marrying Tzeitel. After a misunderstanding (Tevye still thinks he wants to marry his daughter), they agree to Tzeitel marrying Lazar Wolf, and drink on it. While they sing (To Life), Russians who were eavesdropping join in the singing. It is the first scene within which the "outsiders" but in on a Jewish moment. After the festivities, when Tevye and Lazar Wolf are stumbling drunk, they run into the Constable, who warns Tevye that a minor pogrom is coming, and that he should warn the others. Tevye is slightly sobered, but clinging to the fortunate match for his eldest daughter.
The next morning, Tevye is having a lay-in after his night of drinking. While Perchik tries to teach the girls the employer is to be distrusted, Golde hustles them to do Tevye's chores, since he is still asleep. Tevye eventually wakes, and goes to tell Tzeitel that she's to be congratulated-- that she's to be married to Lazar Wolf, much to her devestation. When she begs not to be married, he is mostly swayed. Motel, the tailor, makes his move then, and after much begging from the couple, Tevye gives in to their pleas (after lamenting the state of Tradition in his monologue) and gives them permission to marry. Once Tevye gives them some space, Motel, elated, cheers on his own courage for asking Tevye's permission finally, a year after his pledge to Tzeitel that they would marry (in Miracle of Miracles).
Tevye, however, is discontented by this choice, and not sure how to break it to Golde, so they go to bed before the subject is broached. Plagued by bad dreams, he uses it as a way to break the news to her-- by describing the horrors he'd seen in his dream (found in the Dream), including visitations from Golde's mother Tzeitel, and Lazar Wolf's dead wife Fruma-Sarah.
In act 1 scene 8, there is no singing, but there is a development of a few relationships. We see the townspeople gossiping about Tzeitel marrying Motel, not Lazar Wolf, Motel buying the hat Lazar Wolf ordered, since he got his bride (and therefore he can get his hat). Perhaps most importantly, Tevye's daughter Chava interacts with Fyedka the Russian seemingly for the first time. He shuts down his friends' teasing of her, and sends them away so he can try and lend her a book, that they can then discuss. Chava rejects it at first, but eventually accepts it.
After Fyedka and Chava's possibly ill-advised interaction, the scene turns to a joyous occasion, which is Tzeitel and Motel's wedding. Tevye and Golde lament the change of life with Sunrise, Sunset. Lazar Wolf turns up at the wedding to try and show there are no hard feelings when there clearly are. Perchik tries to get a woman to dance with him, which is a challenging idea for the people of Anatevka (read a little about the history of separated dancing in this article about the Hora). Unfortunately, Lazar Wolf isn't the only guest that turns up; mid celebration, the constable and his men turn up to create the mischief previously mentioned. Perchik attempts to fight them, and when he is injured, the constable declares it enough, and they leave. This note is the end of act 1, and signals a shift in the tone of the play.
⋯ACT 2⋯
Act 2 opens with Tevye summarizing the past two months, after Motel and Tzeitel's wedding. The story then immediately moves into Hodel treating Perchik coldly, because he has to return to Kiev, and Hodel doesn't want him to leave. He asks her a political question; to promise to marry him. Much happier, Hodel agrees, and Perchik proclaims (in song) that now he has everything -- more than everything, in fact.
When the couple break the news to Tevye, however, he is much less than pleased, because not only is Perchik leaving Hodel for Kiev with only a promise to send for her to be married there, they aren't even asking Tevye's permission, let alone letting him find a match for Hodel-- they're telling him that it's happening and they merely wish to have his blessing. Tevye's Rebuttal takes place largely in song form, but he works himself into caring more about his daughter's happiness than Tradition, again.
Tevye, thinking about marrying for love instead of because it's a good match, asks Golde if she loves him. This is a sort of turning point for the couple, as they previously seem to have a slightly resentful relationship, and while they say it doesn't change anything, they seem a little more resolved.
Anatevka is wonderful at one thing: Rumors. The next song is all about spreading information and the chain of miscommunication about Tevye's family. Yente blames the family's (slightly exaggerated) misfortune on men and women dancing together, or: the breakdown of tradition.
The truth of the rumors is revealed to be that Perchik was arrested in Kiev, and sent to Siberia, to a prison camp. Hodel tries to explain to her father in this song that she is going to be with him, not because he asked her to, but because that's where she belongs.
The next scene again has no singing. Within it, we see that Motel finally has his sewing machine -- and they have a baby. Additionally, Chava tries to tell Tevye she wishes to marry Fyedka, but the concept of an exogenous marriage is so complicated that Tevye doesn't hear her out and forbids her from seeing Fyedka. The next morning, the consequences are clear; Chava has eloped, married to Fyedka by a priest. Tevye declares her dead to Golde. when Chava requests his acceptance, he has a song lamenting that she's strayed so far from tradition, and rejects her again. This page includes information about Tevye's song, and the tradition of endogenous (within the community) marriage in Jewish culture.
The musical concludes with a bittersweet song to Anatevka, as the constable warns that in just a few days there will be a true pogrom. As was the experience of many Jewish people once from the Pale of Settlement, everyone resolves to leave Anatevka. Tevye, Golde, Shprintze, and Bielke leave for America. Motel, Tzeitel, and their baby plan to join them as soon as possible. Chava and Fyedka have a brief reprieve from Tevye's rejection when they reveal they can't stay in Anatevka, with people who want to hurt Jewish people just for being Jewish. The choice to show the very real effects of the Pogroms were not in the original stories of Tevye der Milkhiker, but added to ground the musical, written in 1964, in reality for the children of those who came from the Pale in the early 20th century.