(Joint post from me and MarGavriel)
I don’t know about you, but when someone says “Selihot,” my heart sinks, because in my experience, selihot are Hebrew Text Walls of Doom, muttered incomprehensibly and far too fast, punctuated by wails of Divine Attributes which are the only bits I actually recognise. Sound familiar?
Apparently (who knew?) when done properly, they’re actually poems with actual meaning. Not just text walls of doom. More on one verse of one of them in just a moment, but first – liturgically, what exactly are selihot?
Selihot are poems originally recited by the cantor, in his repetition of the Amidah. On weekday fasts, they form part of the berakha סלח לנו, and on Yom Kippur, part of the middle berakha, the Yom Kippur one. Before, after, and between the poems, the 13 Attributes of Divine Mercy (ה’ ה’ אל רחום וחנון) are recited, prefaced by either אל ארך אפים or אל מלך יושב.
In recent centuries, almost all our communities have removed the Selihot liturgy from its original context, and placed it after the whole Hazzan’s Repetition, presumably because of concerns of hefsek [thought-train derailment]. Some few communities resist the urge to destroy, and retain the original structure; if yours does, feel free to leave a note in the comments for the edification of others.
In recent years, communities have also removed the Selihot liturgy from the prayerbook and placed it instead on grubby photocopied handouts, but you can find this one (by Solomon ibn Gabirol) in Artscroll, on page 868. Here’s a sound file of the stanza.

גְּדוֹר פִּרְצִי בְּבֶן פַּרְצִי / וּמֵחֶדֶק לְקוֹט שׁוֹשָׁן	Repair my breach with the descendant of Peretz [i.e., the Messiah], / and collect the lilies [Israel] from amidst the brambles.
בְּנֵה בֵּית זְבוּל וְהָשֵׁב גְּבוּל / הַכַּרְמֶל וְהַבָּשָׁן	Build the Temple Dwelling, and restore the borders / of Carmel and Bashan.
וְעַיִן פְּקַח וְנָקָם קַח / מֵאֵצֶר וּמִדִּישָׁן	Keep thine eye alert, and take vengeance / from Etzer and Dishan [Biblical Edomite groups, i.e. Roman-Christians].
שְׁפוֹט אִלֵּם וְאָז יְשַׁלֵּם / הַמַּבְעֶה וְהַמַּבְעִיר	Bring justice to the mute one [the Jewish people], and then / may the destroyer and burner pay back —
יוֹם גָּבַר הָאוֹיֵב וַתִּבָּקַע הָעִיר	The day when the enemy overpowered [us], and the City went under siege.

17 Tammuz, by the way, is the only Minor Fast to be mentioned in the Mishna (m. Taanit 4:6), where it is juxtaposed to 9 Av:

חמישה דברים אירעו את אבותינו בשבעה עשר בתמוז, וחמישה בתשעה באב. בשבעה עשר בתמוז נשתברו הלוחות, ובטל התמיד, והובקעה העיר, ושרף אפסטמוס את התורה, והעמיד צלם בהיכל…

Five things befell our ancestors on 17 Tammuz, and five on 9 Av. On 17 Tammuz, (a) the Tablets were smashed, (b) the Tamid-offering ceased, (c) the City was besieged, (d) Apostomos burned the Torah-scroll, and (e) an idol was set up in the Temple…

This kind of text isn’t unknown in the Mishnah, but it’s perhaps a trifle unexpected. The Mishnah is the realm of legalese, of rulings, of law. Why here does it speak of history, of identity, of nonlegal matters?
The poem’s line שְׁפוֹט אִלֵּם וְאָז יְשַׁלֵּם / הַמַּבְעֶה וְהַמַּבְעִיר (bring justice to the mute one, and then / may the destroyer and burner pay back) is very clever language, when you look at it. In just a few words, the poet invokes huge swathes of Talmudic discourse, all developing very central Jewish ideas of justice and obligation – where people play fair, and bring disputes to the court, and things are settled properly.
But that’s just the problem. Our enemies, whether Titus or anyone else, don’t play fair. And they get away with it. And we can’t judge them in human courts. And it’s beastly unfair.
So we pray to God: שפוט אלם – “give fair judgment to the mute [‘Am Yisra’el], and only then will the מבעה ומבעיר pay up”. Bring the judgements the court would render, if we could only get these people into court.

…בתשעה באב נגזר על אבותינו שלא ייכנסו לארץ, וחרב הבית בראשונה, ובשנייה, ונלכדה ביתר, ונחרשה העיר. משנכנס אב, ממעטין בשמחה.

…On 9 Av, (a) it was decreed that our ancestors would not enter the Land [at the time of the Spies], and the Temple was destroyed (b) the first time, and (c) the second time, and Bethar was captured [by the Romans, from Bar Kosiva’s insurgents, in the year 135], and (e) the City was plowed [to utter destruction]. Once the month of Av enters, we decrease our joy.

So these are the Three Weeks of Doom, starting now and culminating on 9 Av, in the destructions of Jewish direction, spirituality, hope, pride, identity. This is the time of year when we remind ourselves what it is like to have nothing.
Nothing save what’s inside. The voice of the poet, calling from the brambles, praying for God to bring us justice. “God – we are Jews, and we try to play by the rules – the Torah’s שלם ישלם המבעיר את הבערה and Bava Kamma’s ארבעה אבות נזיקין: השור והבור והמבעה וההבער and that sense of fairness and justice is part of what makes us Jewish. Take that away, and we are disoriented unbearably. Restore that. Please.”