Maimonides summarizes the earlier sources and the Talmudic discussions upon them as follows:
Mishneh Torah, Book of Adoration, Laws of Prayer and the Priestly Blessing, Chapter 3, Halacha 1
תְּפִלַּת הַשַּׁחַר: מִצְוָתָהּ שֶׁיַּתְחִיל לְהִתְפַּלֵּל עִם הָנֵץ הַחַמָּה; וּזְמַנָּהּ עַד סוֹף שָׁעָה רְבִיעִית — שֶׁהוּא שְׁלִישׁ הַיּוֹם, וְאִם עָבַר אוֹ טָעָה וְהִתְפַּלֵּל אַחַר אַרְבַּע שָׁעוֹת עַד חֲצוֹת הַיּוֹם — יָצָא יְדֵי חוֹבַת תְּפִלָּה, אֲבָל לֹא יָצָא יְדֵי חוֹבַת תְּפִלָּה בִּזְמַנָּהּ; שֶׁכְּשֵׁם שֶׁמִּצְוַת תְּפִלָּה מִן הַתּוֹרָה, כָּךְ מִצְוָה מִדִּבְרֵיהֶם לְהִתְפַּלֵּל אוֹתָהּ בִּזְמַנָּהּ שֶׁתִּקְּנוּ לָהּ חֲכָמִים וּנְבִיאִים.
The morning prayer: Its Mitzvah is to begin praying with sunrise; and its time frame is until the end of the fourth time-unit which is a third of day, and if that [time] passed, or he erred, and he prayed after four time-units until midday — he has fulfilled his obligation to pray, but did not fulfill his obligation to pray in its [proper] time; just as it is a Mitzvah to pray according to the Torah, so it is a Mitzvah according to their teachings (— the Sages) to pray it (— that prayer) in its [proper] time that was established for it by Sages and Prophets.
Mishneh Torah, Book of Adoration, Laws of Prayer and the Priestly Blessing, Chapter 3, Halacha 8
כָּל מִי שֶׁעָבַר עָלָיו זְמַן תְּפִלָּה וְלֹא הִתְפַּלֵּל: בְּמֵזִיד — אֵין לוֹ תַּקָּנָה וְאֵינוֹ מְשַׁלֵּם, בְּשׁוֹגֵג אוֹ שֶׁהָיָה אָנוּס אוֹ טָרוּד — מְשַׁלֵּם אוֹתָהּ תְּפִלָּה בִּזְמַן תְּפִלָּה הַסְּמוּכָה לָהּ, וּמַקְדִּים תְּפִלָּה אַחֲרוֹנָה, וְאַחֲרֶיהָ מִתְפַּלֵּל אֶת הַתַּשְׁלוּמִין.
Whomever the time for prayer passes by him, and he did not pray: Intentionally — there is no resolution, and he cannot compensate [for it], By accident or by circumstances beyond his control or was busy — he compensates that prayer at the time of the upcoming prayer, and he precedes with the latter prayer and follows it with the compensatory prayer.