Gemara

Mishnah - R. Gamliel: Anyone who did not say (why we eat) the following three things was not (properly) Yotzei - Pesach, Matzah and Maror.

We eat Pesach because Hash-m skipped over (or had mercy) on the houses of our ancestors in Mitzrayim (when He killed their firstborn) - "Va'Amartem Zevach Pesach Hu la'Shem Asher Posach..." ("Va'Amartem" commands to tell about Pesach; Matzah and Maror are equated to Pesach.)

We eat Matzah to commemorate the redemption of our ancestors from Mitzrayim - "Va'Yofu Es ha'Batzek Asher Hotzi'u mi'Mitzrayim (Ugos Matzos)".

Abarvenel (Pirush ha'Hagadah): Matzah shows the speed of the redemption. Normally, the journey to Sukos should take a day (Rashi Shemos 12:37 - it is 120 Mil, a three days journey), and Yisrael arrived before the dough fermented, i.e. within the time to walk one Mil!

We eat Maror to commemorate how the Mitzriyim embittered the lives of our ancestors in Mitzrayim - "Va'YMaRaRu Es Chayeihem".

(Rava): One must lift the Matzah and Maror for everyone to see (when he says 'this Matzah that we eat' and 'this Maror...' Nowadays, if one or both of the cooked foods are meat,) one need not lift the meat;

Moreover, one should not do so, for it looks like he eats Kodshim outside the Mikdash (Rashbam - it looks like he makes the meat the Hekdesh)!

Rishonim

Rif and Rosh cite the Mishnah and Rava verbatim.

Rambam (Hilchos Chametz u'Matzah 7:5): Anyone who did not say Pesach, Matzah and Maror was not Yotzei. All of these are called Hagadah.

Rambam (ibid. 8:4): He says 'this Pesach that we eat is because Hash-m Posach on the houses of our ancestors in Mitzrayim - "Va'Amartem Zevach Pesach..." He lifts the Maror in his hand and says 'this Maror that we eat is because the Mitzriyim embittered..."Va'YmaRaRu...' He lifts the Matzah in his hand and says 'this Matzah that we eat is because the dough of our ancestors did not rise before Hash-m appeared to them and redeemed them - "Va'Yofu..."

R. Mano'ach: The Rambam agrees that we explain Matzah before Maror. He merely copies his text of the Mishnah, which put Maror first, because the embittering preceded the redemption.

Poskim

Shulchan Aruch (OC 473:7): One reads the Hagadah. When he reaches 'this Matzah', he must lift the Matzah to show it to everyone seated to endear the Mitzvah to them. One should lift the broken Matzah, which is like Lechem Oni. He does similarly when he reaches 'this Maror'.

Darchei Moshe: If one did not mention Pesach, Matzah and Maror, b'Di'eved he was Yotzei Hagadah.

Maharsha (DH Kol): One must say them to be Yotzei eating them.

Mishnah Berurah (64): Hagadah is obligatory also for women. A servant should be at the table to hear the entire Hagadah. If she must cook, at least she must hear the explanations of Pesach, Matzah and Maror, for otherwise she is not Yotzei.