Astute readers will surely recall my writing regarding tachanun. I decided to take a peak at the relevant halachos again, and I'm happy to present, for your edification, the laws of Tachanun II: Knowledge is Power.
Halacha 1: After the Chazzan finishes the repetition of the amidah the congregation falls on their faces and requests mercy (ומתחננים), every place according to its custom. The whole concept of tachanun is only a minhag that Israel had the custom of doing from the earliest days, and nevertheless it's not an obligation, it's a matter of choice (רשות). Therefore all it's halachos are dependent on minhag, as will be explained.
Tachanun must be said immediately following tefilah (shemone esrei), and if a person pauses and does other things after tefilah (before tachanun) then his tachanun is not accepted as much. Therefore a person should not talk between shemone esrei and tachanun (even though according to Halacha there's no prohibition here, since tachanun is רשות). We are only worried here about doing entirely different things, but a little speech is allowed. Anyone who doesn't pause to deal with extraneous matters at all can go to another place and fall there on his face.
The main falling upon the face (נפילת אפים) that was done from the earliest days is to fall upon one's face on the ground, even without placing there his hands and feet (including sitting on the ground and falling on his face). It isn't fitting for an important person to behave like this when he's in front of (davening with) the congregation if the congregation doesn't do it also (i.e. don't try to be too frum) unless he is so confident in his deeds and knows with certainty that he will be answered like Yehoshua Bin Nun, as it says (about him), "Raise yourself up, why do you fall on your face?" (meaning that he was answered immediately by Hashem). This is because if he isn't answered immediately then the congregation will wonder about him, saying that he isn't ready or fit to be answered.
Similarly the rabbis decreed that every person is forbidden to prostrate himself on a stone floor, even if he doesn't spread out his hands and feet, because we are worried that he will completely prostate himself there, which would be a violation of the Torah (and for which he is whipped), as the verse says, "And you shall not emplace a flooring of stone upon which to prostrate oneself," as this is only permitted in the Beis Hamikdash.
Falling on the ground is forbidden, but placing your face on your side is permitted, even for an important person in front of the congregation, and even on a stone floor, without putting the hands and feet down, as was explained. In these days, when we don't actually fall upon our faces but rather put our heads down and cover our faces, none of this is applicable.
Therefore it's permitted to say tachanun even while standing, but according to Kabbalah a person should say it while sitting. If it's impossible to say it sitting while within the three steps that were taken following the shemone esrei then a person should wait the amount of time necessary to walk four amos and then return to his place and say tachanun there while sittting.
Our custom is to cover our faces with a garment, and it's not enough to merely cover with the arm upon which we fall, since the arm and the face our part of one body, and it's not possible for something to cover itself, as was explained in hilchos krias shema and shemone esrei.
The custom in many places is to lean on the left side because when a person davens the Shechinah is on his right side, as the verse states, "Hashem is your protective Shade at your right hand," and as was explained in the laws of shemone esrei. When a person rests on his left side then he is facing the right (the Shechinah), as opposed to if he leaned on the right side, in which case his back would be facing the shechinah, which would be improper, because it's not fitting for a servant to turn his back to his master.
There are those who say that it's proper to lean on the right side, because when you lean on the right side the Shechinah is opposite you, and then he will have in mind, "His left hand is under my head, and his right arm embraces me."
In these lands we have the custom to lean on our right arms by shacharis because of the honor due to the tefillin which are on the left arm. By mincha (and so too by shacharis when tefillin aren't worn on the left arm) he should lean on his left arm. The chazzan, who is standing on the right side of the ark, should tilt his his head a little even when he falls on his right side (and this is also the din of those who sit next to the menorah before the ark who lean on their faces facing east like the chazzan).
According to Kabbalah the psalm אליך ה׳ נפשי אשא (Psalms 25) should be said. One who says this and his heart is far (i.e. he is not on the level) will cause himself to die before his time. Therefore in this country we have the custom to say רחום וחנון כו׳ ה׳ אל באפיך כו׳.
(In the Siddur the Alter Rebbe says to say both.)
Halacha 2: After a person does נפילת אפים he should lift up his head and ask for mercy a bit, while sitting in his place, every place according to their custom. The basic custom is to say ואנחנו לא נדע מה נעשה וכו׳ since we davened on everything that a person is able to daven, whether sitting, standing, or leaning. This is what Moshe Rabbeinu A"H did, as the verse says, "And I sat on the mountain," and, "Then I threw myself down before Hashem." After this, since we no longer have the strength to daven in any other way, we say, "And we do not know what to do etc," and afterwords we say half-Kaddish, Ashrei, and Lamnatzeiach.
Halacha 3: There are those who have the custom to only do נפילת אפים in the presence of a Sefer Torah, and a source for this can be found in the war of Ai (Joshua 7, 6) as it says, "And [Joshua] fell on his face to the ground before the ark of Hashem." In a place where there is no Sefer Torah the custom is to say tachanun without covering the face.
The custom is to do נפילת אפים in the courtyard of a shul when the courtyard is open to the shul itself (and people daven there when the shul is open) as is done in the shul itsef. However, when part of the shul is closed off from the part that contains the Sefer Torah then the custom is not do to נפילת אפים.
The women's shul is considered to be separate in this matter from the shul with a Sefer Torah inside, and if it's not open the custom is to do נפילת אפים. However, the courtyard outside the shul has the law of a courtyard inside a shul-if it's open, you do, if it's not open, you don't.
Even one Jew davening on his own, if he davens at the same time as the congregation davening in the shul, is able to say tachanun while doing נפילת אפים. This is because he is considered to be standing with them, and even an iron curtain can not separate between the Jewish people and their Father in Heaven when there are ten Jews present, as explained in the laws of the morning blessings.
Halacha 4: There is no נפילת אפים at night, because of the known reason (to those who know), and therefore if someone postponed davening mincha until the night he shouldn't do נפילת אפים. Therefore one should skip אבינו מלכנו in order todo נפילת אפים in the day; nevertheless, one should only do this when it's certainly night, but not if it's שקיעה. The custom is to do נפילת אפים on the night of selichos since it's close to the day anyway (the chabad custom is not to do this). There are those who have the custom to lengthen selichos until it's definitely daytime, and then they do נפילת אפים.
Part 2 somewhen...