The Rambam discusses the idea of the kos chamishi, the fifth cup, in the eighth chapter of Hilkhot Chametz uMatza.
ואחר כך נוטל ידיו, ומברך ברכת המזון על כוס שלישי, ושותהו. ואחר כך מוזג כוס רביעי, וגומר עליו את ההלל; ואומר עליו ברכת השיר, והיא "יְהַלְלוּךָ יְהוָה אֱלֹהֵינוּ כָּל מַעֲשֶׂיך . . . " ואינו טועם אחר כך כלום, כל הלילה--חוץ מן המים. ויש לו למזוג כוס חמישי, ולומר עליו הלל הגדול--והוא מ"הודו לה', כי טוב" (תהילים קלו,א), עד "על נהרות, בבל" (תהילים קלז,א); וכוס זה אינו חובה, כמו ארבע הכוסות. ויש לו לגמור את ההלל בכל מקום שירצה, אף על פי שאינו מקום סעודה.
And afterward he washes his hands and says birkat haMazon on the third cup and drinks it. And afterward he pours the fourth cup and completes upon it the hallel - and he says birkat haShir on it, which is, "They shall proclaim Your praise, Hashem, our God - all Your works...", and he may not taste anything else the rest of the night -- except for water. He may pour a fifth cup, and say on it hallel haGadol, the Great Praise, which is from, "Give thanks to Hashem for He is good", until, "By the rivers of Babylon" [meaning, all of Tehillim 136]. This cup is not a requirement like the four cups. He may complete the hallel wherever he so chooses, even if it is not the place of the meal.
For a more detailed background on this sugya I recommend this article written by Rav Ezra Bick and Rav Yair Kahn. I would like to offer an interpretation of the purpose of the optional fifth cup.
The Rambam explains the purpose of the four cups in the seventh chapter of Hilkhot Chametz uMatza:
ח [ו] בכל דור ודור, חייב אדם להראות את עצמו כאילו הוא בעצמו יצא עתה משיעבוד מצריים, שנאמר "ואותנו, הוציא משם . . ." (דברים ו,כג). ועל דבר זה ציווה בתורה, "וזכרת, כי עבד היית" (דברים ה,יד; דברים טו,טו; דברים טז,יב; דברים כד,יח; דברים כד,כב), כלומר כאילו אתה בעצמך היית עבד, ויצאת לחירות ונפדית.
ט [ז] לפיכך כשסועד אדם בלילה הזה, צריך לאכול ולשתות והוא מסב דרך חירות. וכל אחד ואחד, בין אנשים בין נשים, חייב לשתות בלילה הזה, ארבעה כוסות של יין--אין פחות מהן; ואפילו עני המתפרנס מן הצדקה, לא יפחתו לו מארבעה כוסות: שיעור כל כוס מהן, רביעית.
In every generation I man is obligated to present himself as if he himself went out at that moment from the bondage of Egypt, as it says, "and us - He took out from there..."(D'varim 6:23). And concerning this matter the Torah commands, "And remember, that you were a slave" (D'varim 5:14; 15:15; 16:12; 24:18; 24:22), meaning, it is as if you yourself were a slave and you went out to freedom and were redeemed.
Therefore, when a person dines on this night, he needs to eat and drink while he is reclining in the way of freedom. And every single person, male and femal, is obligated to drink on this night, four cups of wine, no less; and even a poor person who is supported by charity, they should not give him less than four cups. The required quantity of each cup is a reviit.
Drinking the four cups is a fulfillment of the obligation to view oneself as having experienced the redemption. The goal of the Seder is to behold God as גאל ישראל, the Redeemer of Israel. To do this we must ourselves feel redeemed. We say hallel to demonstrate that we have experienced redemption.
In "The Joy of My Heart", I wrote the following concerning the general topic of Sefer Z'manim:
The choice of "נָחַלְתִּי עֵדְוֹתֶיךָ לְעוֹלָם: כִּי-שְׂשׂוֹן לִבִּי הֵמָּה." to crown Sefer Z'manim is perfect. It both captures the joyous spirit of the yamim tovim and the sorrow of the ששון, joy, not yet attained because (Eikha 5:2) " נַחֲלָתֵנוּ נֶהֶפְכָה לְזָרִים, בָּתֵּינוּ לְנָכְרִים. " - " Our heritage has been turned over to strangers, our houses to aliens. ".
This lack is beautifully expressed in the b'rakha we say on the completion of Magid:
בָּרוּךְ אַתָּה יְהוָה אֱלֹהֵינוּ מֶלֶךְ הָעוֹלָם, אֲשֶׁר גְּאָלָנוּ וְגָאַל אֶת אֲבוֹתֵינוּ מִמִּצְרַיִם, וְהִגִּיעָנוּ לַלַּיְלָה הַזֶּה לֶאֱכֹל בּוֹ מַצָּה וּמְרוֹרִים. כֵּן יְהוָה אֱלֹהֵינוּ וֵאלֹהֵי אֲבוֹתֵינוּ, יַגִּיעֵנוּ לְמוֹעֲדִים וְלִרְגָלִים אֲחֵרִים הַבָּאִים לִקְרָאתֵנוּ לְשָׁלוֹם, שְׂמֵחִים בְּבִנְיַן עִירָךְ וְשָׂשִׂים בַּעֲבוֹדָתָךְ, וְנֹאכַל שָׁם מִן הַזְּבָחִים וּמִן הַפְּסָחִים שֶׁיַּגִּיעַ דָּמָם עַל קִיר מִזְבְּחָךְ לְרָצוֹן, וְנוֹדֶה לְךָ שִׁיר חָדָשׁ עַל גְּאֻלָּתֵנוּ וְעַל פְּדוּת נַפְשֵׁנוּ. בָּרוּךְ אַתָּה יְהוָה, גּוֹאֵל יִשְׂרָאֵל.
Blessed are you Hashem our God King of the Universe, who redeemed us and redeemed our forefathers from Egypt, and enabled us to reach this night to eat matza and marror. So Hashem our God and the God of our forefathers should enable us to reach other convocations and festivals that are coming to greet us in peace, happy in the building of Your city and sasim, joyful in your service, and we shall eat there from the sacrifices and from the paschal offering that their blood should reach the wall of Your altar for favor, and we shall give thanks to You a new song for our redemption and on the saving of our souls. Blessed are you Hashem, Redeemer of Israel.
The lack of the ultimate redemption holds us back from being able to give the highest praise. However, this limitation is not absolute. The seder night can be so revelatory to an individual that he is able to experience the true sasson of the festival. The fifth cup represents the possibility of a personal redemption that transcends the limitations of being in exile. The hallel haGadol said on the fifth cup is none other than the kol sasson of the y'mot haMashiach, the Messianic times:
Yirmiyahu 33:
י כֹּה אָמַר יְהוָה, עוֹד יִשָּׁמַע בַּמָּקוֹם-הַזֶּה, אֲשֶׁר אַתֶּם אֹמְרִים, חָרֵב הוּא מֵאֵין אָדָם וּמֵאֵין בְּהֵמָה--בְּעָרֵי יְהוּדָה, וּבְחֻצוֹת יְרוּשָׁלִַם, הַנְשַׁמּוֹת מֵאֵין אָדָם וּמֵאֵין יוֹשֵׁב, וּמֵאֵין בְּהֵמָה. יא קוֹל שָׂשׂוֹן וְקוֹל שִׂמְחָה, קוֹל חָתָן וְקוֹל כַּלָּה, קוֹל אֹמְרִים הוֹדוּ אֶת-יְהוָה צְבָאוֹת כִּי-טוֹב יְהוָה כִּי-לְעוֹלָם חַסְדּוֹ, מְבִאִים תּוֹדָה בֵּית יְהוָה: כִּי-אָשִׁיב אֶת-שְׁבוּת-הָאָרֶץ כְּבָרִאשֹׁנָה, אָמַר יְהוָה. {ס}
10. So said the Lord: There shall again be heard in this place, concerning which you say, "It is desolate without man and without beast," in the cities of Judah and in the streets of Jerusalem that are desolate without a man and without an inhabitant and without a beast, 11. the sound of mirth and the sound of joy, the voice of a bridegroom and the voice of a bride, the sound of those saying, "Thank the Lord of Hosts, for the Lord is good, for His loving-kindness endures forever," bringing a thanksgiving offering to the House of the Lord, for I will restore the captivity of the land as at first, said the Lord.
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