Gemara

Question: Why does it say "they went up to the south, and he came to Chevron"? It should say 'they came'!

Answer (Rava): Kalev separated from the counsel of the Meraglim, and came alone to the graves of the Avos, to ask the Avos to request mercy for him, lest he accede to the Meraglim's counsel.

Ta'anis 16a - Question: Why do we go out to the cemetery on a Ta'anis (in the series of seven fasts due to lack of rain)?

Answer #1 (Levi bar Chama or R. Chanina): We show that we are considered like Mesim in front of Hash-m.

Answer #2 (the other of Levi bar Chama and R. Chanina): We ask the dead to request mercy for us.

Bava Metzi'a 85b (R. Chaviva): Eliyahu used to speak with a certain Chacham. In the morning, his eyes looked fine. In the evening, they looked singed.

The Chacham: Eliyahu permitted me to look at all Chachamim going to the Heavenly academy, except for the chair of R. Chiya.

The Chacham could not hold himself back; he looked. Two fiery sparks blinded his eyes. The next day, he prostrated near the cave in which R. Chiya was buried, and mentioned that he learns the Tosefta of R. Chiya (so R. Chiya would be more inspired to pray for him) - he was healed.

Rishonim

Rif and Rosh (Ta'anis 5b and 2:1): Levi bar Chama and R. Chanina argued about whether we go out to the cemetery on a Ta'anis to show that we are like Mesim in front of Hash-m, or to ask the dead to request mercy for us.

Rambam (Hilchos Ta'anis 4:18): In the set of seven fasts due to lack of rain, after Tefilah everyone goes to the cemetery and supplicates there, to show that 'you will die like them if you do not repent.'

Poskim

Shulchan Aruch (OC 559:10): On Tish'ah b'Av, we go to the cemetery immediately after leaving the Beis ha'Keneses.

Magen Avraham (15): Also the dead are in pain (over the Churban). Therefore, it does not suffice to go to Nochri graves. However, if there are no graves of Yisrael, we go to Nochri graves.

Mishnah Berurah (41): This is so the Mesim will request mercy upon us.

Kaf ha'Chayim (81): R. Chaim Vital once went to a Nochri grave, and it (the Mes) tried to kill him. Some forbid going to Nochri graves due to "Doresh El ha'Mesim" (seeking to bring Ru'ach ha'Tum'ah upon oneself).

Shulchan Aruch (579:3): In the set of seven fasts due to lack of rain, after Tefilah everyone goes to the cemetery and cries and supplicates there, to show that 'you will die like them if you do not repent.'

Question (Gra): Why do the Rambam and Tur rule like the first opinion?

Rema (581:4): (On Erev Rosh Hashanah) in some places the custom is to go to graves and say many supplications and give Tzedakah to the poor.

Darchei Moshe (4 DH Kosav Maharil, citing the Maharil): On Erev Rosh Hashanah and Erev Yom Kipur, the custom is to go to graves to humble oneself and say supplications there, for while his heart is humbled his prayer is accepted. We find that Kalev prostrated on the graves of the Avos.

Be'er Heitev (17 citing Maharil, and Machatzis ha'Shekel DH Kivrei, Kitzur Shulchan Aruch 128:13, Mishnah Berurah 27, Kaf ha'Chayim 95): A cemetery is a Kadosh and Tahor place. We do not put our hope in the Mesim, rather, that Hash-m will answer us in the merit of the Tzadikim buried there.

Kitzur Shulchan Aruch: If one puts his hope in the Mesim, this is close to Doresh El ha'Mesim.

Eshel Avraham (15): Ma'aneh Lashon brings Tefilos to say on the graves. It seems that we request them to advocate for us.

R. Chayim Palti'el (brought in Bach YD 217 b'Sof DH Mi): A vow to go to the cemetery is like Doresh El ha'Mesim (the Isur to seek communication with the dead). Kalev went to Chevron to prostrate on the graves of Tzadikim to help his Tefilah be received. If one disgraced a person and he died, it is proper to go to his grave and prostrate on it, to show honor. One who does not have one of these intents has no reason to go. I help him to annul his vow.

Rebuttal (Bach): The custom is for people to go, and no one protests. The Zohar supports this. He should pray that Hash-m help him in the merit of his ancestors.

Igros Moshe (OC 5:43): Poskim argue about whether or not one may request from angels that they fulfill their missions in the best way for us. All forbid to request from them anything else. A Mes cannot pray, but it is possible for his Neshamah to pray. Seemingly, the Neshamah is like an angel without any mission, so all the more so we may not request from it. Mesim are exempt from Mitzvos, so we cannot ask them to pray for us. However, their children or others can ask Hash-m to answer prayer in the merit of Mesim, for the dead are pained by the pain of their children or others they knew, and certainly by the pain of Kelal Yisrael. This is a reason to be more lenient about asking Mesim. We arouse them to pray for us to alleviate their own pain. Also, perhaps they are more sensitive to human pain than angels are. The Or Zaru'a holds like the first side; he holds that Doresh El ha'Mesim includes requesting from the dead. The Bach permits, like the second side. He holds that Doresh El ha'Mesim is only doing actions to bring Ru'ach ha'Tum'ah on himself.

B'Tzel ha'Chachmah (4:30): R. Yehudah ha'Chasid said that one who lives in a city with a cemetery and goes to another city should not pray to the dead there, lest the dead in his city feel insulted and harm him. People are not careful about this, and leave Kadosh cities like Yerushalayim to pray by graves of Tzadikim elsewhere. It seems that R. Yehudah did not discuss one who goes to pray by the grave of a particular Tzadik, relative, Rebbi... We may infer that R. Yehudah ha'Chasid permits praying to Mesim, i.e. that they should advocate for him to Hash-m. The Be'er Heitev and others cite the Maharil, who forbids. However, others prove from the Gemara that it is a Mitzvah to request Mesim to advocate for us. Yosef Ometz brings a Tefilah from R. Tam asking angels to pray for us.