Gemara

(Rav Chanah bar Bisna): One who prays must picture the Shechinah in front of him - "Shivisi Hash-m l'Negdi Tamid."

Berachos 28b (Mishnah - R. Eliezer): If one makes his prayer Keva (fixed), it is not considered supplications.

29b - Question: What does this mean?

Answer #1 (R. Oshaya): It is like a burden to him.

Answer #2 (Rabanan): He does not say it like supplications.

Answer #3 (Rabah and Rav Yosef): He is unable to (concentrate to) request something new.

R. Zeira: I could request something new, but I fear lest I would get confused (and be unable to resume Shemoneh Esreh).

31a (Rav Hamnuna): "V'Chanah Hi Medaberes Al Libah" teaches that one who prays must have intent.

Rishonim

Rif (Berachos 20a): R. Oshaya says that Keva is when prayer is like a burden to him. Rabanan say that it is when he does not say it like supplications.

R. Yonah (DH Gemara): Rav Hai Gaon says that even though Rabanan say that even though prayer is like a burden to him and he does not say it like one who needs something, rather, due to his obligation, since he says it gently, it is like one who requests mercy for himself. It seems that Rabanan are more stringent in one way, and more lenient in another. Even though prayer is like a burden to him, since he says it one who requests mercy, it is prayer. They are stringent about one who requests mercy for himself, but he does not say it like supplications. R. Hoshaya holds oppositely. One may never pray as if it is a burden, but he need not say it like supplication.

Beis Yosef (OC 98 DH va'Yispalel): The Tur holds that both opinions require both of these l'Chatchilah.

Bach (OC 98 DH v'Nir'eh): The Tur understands that both of these are the same. One who needs something does not rush, rather, he requests gently, like supplication.

Rosh (Berachos 4:18): R. Chananel says that Rabanan require only that he prays like supplications. He need not pray like one who needs something.

Rambam (Hilchos Tefilah 4:16): Kavanah in prayer is to remove all distracting thoughts and consider that he stands in front of the Shechinah. Therefore, one must sit a little before prayer to direct his heart, and then pray.gently, like supplication. He should not make his prayer like a load that he is carrying, and he casts it off and leaves. Therefore, one must delay a little after Tefilah before leaving.

Lechem Mishneh: The Rambam does not bring the opinion of Rabah and Rav Yosef, that one must request something new, for R. Zeira feared to do so.

Poskim

Shulchan Aruch (OC 98:1): One who prays must think in his heart the meaning of the words he says, and consider that the Shechinah is in front of him, and remove all distracting thoughts until his thoughts and intent are clear. If he would speak in front of a mortal king, he would arrange his words and have proper intent, lest he stumble. All the more so in front of the King of kings, who knows all thoughts, he should do so! Chasidim and Anshei Ma'aseh used to isolate themselves and have intent in prayer until they removed physicality and intellect overpowered, until they reached close to the level of prophecy. If another thought comes during prayer, he is quiet until it is Batel. One must think about things that humble the heart and direct it to one's Father in Shamayim, and not think about lightheaded matters.

Source: Orchos Chayim (cited in Beis Yosef DH Kasuv b'Orchos Chayim).

Magen Avraham: The Zohar says that if one comes to unify Hash-m's name, but does not have proper Kavanah and fear, his prayer is cast outside and it is shameful. Therefore, one should not intend for names of Hash-m and Yichudim unless he knows that he will have proper Kavanah,

Mishnah Berurah (1): After R. Shimshon of Kinon learned secrets of Kabalah, he prayed like a one-day old baby. The Pnei Yehoshua says that one cannot think about these intents (in the Shulchan Aruch) at the time of prayer. Rather, he has them in mind before prayer. At the time of prayer he thinks about the meaning of the words.

Kaf ha'Chayim (3): One may not pray with deep Kavanos, e.g. of the Ari Zal, unless he understands Kabalah well.

Kaf ha'Chayim (2): One must be careful also to say the words, letters and vowels properly. The Zohar says that tens of thusands of higher worlds depend on every letter. If one accustoms himself to saying each word properly, like one who counts coins, (later) he will be able to say them slightly quickly. One must be careful not to swallow letters at the beginning of end of words. One must be especially careful to say names of Hash-m with Kedushah, Taharah, and great Kavanah. Hash-m helps one who seeks to purify himself.

Kaf ha'Chayim (5): One who prays with stray thoughts brings a Ru'ach ha'Tum'ah upon himself, and all the more so if he is the Shali'ach Tzibur. Tefilah with Kavanah is a sign that he is a Tzadik. Being as concerned for another's money as for his own is a merit to pray with Kavanah.

Kaf ha'Chayim (11, citing Sefer Chasidim 158): If one has Kavanah only in Berachos discussing his own needs (e.g. Refu'ah or income), this arouses accusations against his prayer.

Rema: One should think before prayer about the exaltedness of Hash-m and man's lowliness, and remove all physical pleasures from his heart.

R. Yonah (22b DH Tzarich): Before prayer one should think about the greatness of Hash-m and the lowliness of man, and remove from his heart wordly pleasures. This will make his Tefilah acceptable.

Shulchan Aruch (3): Prayer must be like a poor person supplicating softly at the door. It should not be like a burden that one seeks to be relieved of.

Tur: If one tries to remove distractions, and prayer is not like a burden to him, he is guaranteed that his Tefilah will be accepted.

Mishnah Berurah (8): One should put to his heart that no created being, angel, Mazel or star has power to fulfill his request, only if Hash-m wants.

Sha'arei Teshuvah: If one prayed without Kavanah for the words in the first Brachah, he was not Yotzei (Shulchan Aruch 101:1). The Rema says that he does not pray again, lest also in the repetition he no have Kavanah. Here is different, for if one prayed unlike supplication, it is easy to pray again like supplication.

Bi'ur Halachah (DH Yispalel): If one prayed to fulfill his obligation, without feeling that he requests his needs, many Poskim require praying again. Their proof is not clear to me. B'Di'eved, one should not pray again, but l'Chatchilah he must be very careful about this. Even those who are stringent are stringent only (if he prayed with an improper attitude) in Birkas Avos.

Kaf ha'Chayim (18, from Sha'arei Teshuvah 66:13): When one says 'who helps the poor', he should consider himself like an Oni begging at the door. Even if he is rich, he should consider himself poor in Torah and Mitzvos. Also, in the grave he will be poor. Through this, his Tefilah is included with that of the poor.

Kaf ha'Chayim (20): It is a terrible sin to rush the Shali'ach Tzibur. The Tefilah must be said gently, and it disrupts his Kavanah.

Magen Avraham (107:4): The Rif's text said that prayer is supplication only if he can be Mechadesh (request something new). This is required only for a Nedavah (voluntary prayer).

Einayim l'Mishpat (Berachos 29b:4): He must explain that Rabah and Rav Yosef discuss only a Nedavah. Why didn't the Rif bring this here? Also, it is difficult to say that a Nedavah must be supplication more than a Nedavah.