How to Put the Face Down
Tilting the Head
When putting the face down today, one should not lay his head straight down on his arm. He should put his elbow on the table with his arm in the air diagonally and then rest his head on his arm such that his face is facing to the side, not completely downward, and covered by his arm.
The reason is that back when they used to actually prostrate themselves on the floor, they had to lean their bodies to the side to avoid the issur of bowing on a stone floor (שו"ע ס"ח ). Even though today we put our faces down without actually bowing, we still fulfill this aspect – that the head should be tilted to the side – to remember the concept (ביאה"ל ד"ה להטות ). Some hold this is halachically required (מג"א סק"כ ).
Covering with a Garment
When putting the face down on the arm, the face must be covered by a garment, not the arm itself. This is because the arm and face are part of the same body, and a body cannot cover itself [because מין במינו אינו חוצץ] (מג"א סק"ב, מ"ב סק"ג ). [If one davens with short sleeves ... and has no garment on his arm as a barrier, he should find some cloth, tissue, or tallis to cover his arm when putting his face down.]
Right Arm or Left Arm?
Most poskim hold it is better to put the face down on the left side. This way, the head tilts to the right, and the Shechina is on a person’s right side when he davens, as the posuk says, “ה' צלך על יד ימינך ” (שו"ע סי' קל"א ס"א, ומ"ב סק"ד, הגר"א ).
Others say one should always put his head down on the right side (י"א ברמ"א ס"א ) since the Shechina is opposite the person, and this way, the Shechina’s left is opposite the person’s right (מ"ב סק"ה ).
Without tefillin. Since most poskim hold it is better to do it on the left side, we do so when there is no reason to do otherwise. This is what we do when one is not wearing tefillin, e.g., at Mincha or if one is not wearing tefillin at Shacharis.
With tefillin. However, when one is wearing tefillin, he should not do it on his left arm out of respect for his tefillin. He should put his face down on his right arm as a sort of compromise since some say to always do it on the right (רמ"א ס"א, ט"ז, מג"א ). This is the prevalent minhag (מ"ב שם ).
However, some hold one should always put his face down on the left, even when wearing tefillin, since the Shulchan Aruch’s opinion is the main one (הגר"א הוזכר במ"ב סק"ו, וע"פ ספרי קבלה כף החיים סק"ל ).
Some put the face down on both arms simultaneously to satisfy both opinions. Since it is clear he is doing it to satisfy both opinions, it is not degrading to his tefillin (שו"ת אג"מ או"ח ח"ה סי' כ' אות י"ט, שו"ת רבבות אפרים ח"ג סי' פ"ו ). When wearing tefillin at Shacharis, one should put his right arm over his left; at Mincha, he should put his left arm over his right (א"א בוטשאטש סוף הסי').
Lefty. At Shacharis, a lefty should put his face down on his left arm, as it does not have tefillin (פמ"ג משב"ז סי' קל"א סק"ב ). He should do the same at Mincha (שו"ת באר משה ח"ב סי' ג').
Sitting or Standing?
Vidui and the 13 Middos are specifically said standing; the same is true for והוא רחום (שו"ע סי' קל"ד ס"א ). Some say not standing for these is called a breach [“פורץ גדר ”] (שו"ע ומ"ב שם ) unless one is unwell.
Putting the head down standing. Some say one may say the part of Tachanun with his face down while standing (ריב"ש הובא במ"ב סק"י ).
Sitting. However, the Shulchan Aruch rules it should be specifically said sitting, not standing (שו"ע ס"ב ) based on Kabbala (ב"י ).
No other choice. Thus, with no other choice one may rely on the opinion that allows it to be said standing. E.g., if one has nowhere to sit or someone is davening Shemoneh Esrei behind him so that he cannot take his three steps back or to the side, he should say this part of Tachanun standing (מ"ב סק"י ).
Leaning. I once heard a suggestion that if one has nowhere to sit, he can lean on something heavily enough that if it were removed, he would fall, as that is also considered like sitting. Thus, if one is next to a wall, he can lean with his back on the wall. One can also lean on a shtender or table with his hands heavily enough that if it were removed, he would fall.
