There is a time-honored practice – dating back many centuries – to read Shir Hashirim on Friday night. Some communities read it before Minha on Friday afternoon, others between Minha and Arbit, and some after Arbit.
The text of Shir Hashirim is, essentially, a love story between a man and a woman. The Seror Ha’mor (Rav Abraham Saba, 1440-1508), in his introduction to Shir Hashirim, bemoans the fact that the Apikorsim (heretics) interpret Shir Hashirim literally, as describing an actual relationship between a man and a woman, Heaven forbid. This profanes Shir Hashirim, which our Sages describe as "Kodesh Kodashim" – the most sacred of all the texts in the Tanach. This book allegorizes the close relationship between Am Yisrael and the Almighty, describing the deep love He feels for us and we feel for Him. One verse in Shir Hashirim says, "Shehora Ani Ve’na’ava" – "I am black but beloved," referring to the fact that Beneh Yisrael are beloved to Hashem even when we’re "black," darkened by sin.
The simplest explanation for why we read Shir Hashirim at the beginning of Shabbat is because Shabbat is like the "wedding" between the Jewish People and G-d. This is why we sing, "Bo’i Kalla" ("Come, O bride") when Shabbat begins, and we dress up in our finest clothing and have an elaborate meal, just like at a wedding. Furthermore, in our Friday night prayer, we refer to Shabbat in the feminine form – "Ve’yanuhu Bah," whereas on Shabbat morning, we use the masculine form – "Ve’yanuhu Bo," and at Minha some have the custom to recite the plural form – "Ve’yanuhu Bam." Some explain these passages as allusions to the bride and groom, who come together on Shabbat afternoon, the time of the "Yihud," the culmination of the "wedding," the height of love and closeness between G-d and His special nation. Appropriately, then, as we celebrate this "wedding," we read Shir Hashirim which tells of the unique feelings of love between G-d and Am Yisrael.
The Hida (Rav Haim Yosef David Azulai, 1724-1806), in his work Kiseh Rahamim, brings another reason. He cites the Seder Ha’dorot as stating that although Beneh Yisrael spent 210 years in Egypt, they worked as slaves for only 117 years. The number 117, then, is associated with the end of exile and suffering, and the onset of redemption, as it was after 117 years of labor that Beneh Yisrael were freed. Our Sages teach us that if we all observe Shabbat properly, then we become worthy of redemption. So, on this day we read Shir Hashirim, which contains 117 verses, as though telling the Almighty that the merit of our observance of Shabbat should bring the end of our exile. Just as G-d redeemed our ancestors from Egypt after 117 years of slave labor, we, too, hope for our redemption in the merit of Shabbat observance, alluded to by the 117 verses in Shir Hashirim. (This is likely the reason why it is customary to read Shir Hashirim after the Seder on Pesach, because of the association between the 117 verses of this book and the Exodus from Egypt.)
Another explanation is based on a teaching of the Roke’ah (Rav Elazar of Worms, Germany, 1176-1238). The Zohar comments that the wicked who suffer in Gehinam are granted a reprieve during the 24 hours of Shabbat. (Our lighting of a candle on Mosa’eh Shabbat signifies the rekindling of the fires of Gehinam.) But in addition, the Zohar writes, the fires of Gehinam cease to burn also when Am Yisrael recites each of the three daily prayers – Shaharit, Minha and Arbit. Specifically, the Zohar says that when Am Yisrael recites one of these prayers, the fires of Gehinam stop burning for an hour and a half – or 4.5 hours each day. Now the six workdays have a total of 144 hours (24 * 6), and among those, the fires of Gehinam are not burning for 27 hours (4.5 * 6). It turns out, then, that the fires of Gehinam burn for a total of 117 hours (144-27). The Roke’ah teaches that King Shlomo composed the 117 verses of Shir Hashirim as a prayer that the merit of this sacred text should save people from the punishment of Gehinam. We therefore recite Shir Hashirim at the end of the week so that the merit of our recitation will save us from the 117 weekly hours of suffering in Gehinam.
Summary: It is customary to read Shir Hashirim on Friday night, either before Minha, after Minha, or after Arbit. The simplest reason is because Shir Hashirim talks of the special love between G-d and Am Yisrael, and Shabbat is the "wedding" between us and Hashem. There are also Kabbalistic reasons for this practice.