One reciting the Amidah must lower his head slightly, so that his eyes point downwards in the way of humility; he must imagine himself standing in the Temple and directing his heart up towards the heavens (Yevamot 105b; Shulchan Aruch 95:2).
The kabbalists praise one who prays with his eyes closed. However, even a person who looks into his siddur follows the law l’chatchilah. Many Acharonim recommend praying from a siddur, so that one can have more kavanah in his prayer (Mishnah Berurah 95:5; Kaf HaChaim 9-10; and see the words of the Ma’amar Mordechai brought in Bei’ur Halachah).
Regarding one’s hands, the Rambam writes (Hilchot Tefillah 5:4) that a person should place his hands on his heart while interlocked, right over left so that he stands as a slave before his master, in awe and fear. That is what the Shulchan Aruch writes (95:3) and what is explained in the Kavanot of the Ari (Kaf HaChaim 95:12). Many maintain that it all depends on the custom of the place; where the Rambam lived, it was, indeed, customary to stand before kings and ministers in the manner in which he described. However, people in other areas practiced differently. For instance, those living in the countries of Edom were accustomed to standing with their arms folded, and those in the land of Ishmael would stand with their hands behind their backs, to symbolically indicate that they do not even have the use of their hands without the consent of the one before whom they are standing (Mahari Abuhav brought by the Beit Yosef; Mishnah Berurah 95:6). Nowadays, according to this, in addition to what the Rambam wrote, it is also permissible to stand with one’s hands adjacent to his body or slightly resting on a shtender (lectern) next to his siddur, for that, too, is considered standing respectfully. However, one should not stand with his hands in his pockets or on his hips, for it is inappropriate to stand that way in front of respected people.
Many people are accustomed to “shuckling” (swaying back and forth) while reciting the Amidah. The Rama writes (Orach Chaim 48; Mishnah Berurah 95:7) that this is the proper way to pray l’chatchilah, in order to express the excitement and trepidation of the prayer experience, and in order to involve one’s whole body in the service of prayer, as it is written (Psalms 35:10), “All my bones will say, ‘Hashem, who is like You.’” By contrast, the Shlah writes that one should not shuckle during prayer, but just the opposite – that standing motionless strengthens one’s kavanah. Furthermore, he says it is not respectful to shuckle. If a person were to come before a human king and begin to shake with his whole body, indeed, the king would immediately dismiss him. If so, one should certainly not act that way in prayer. The Shlah explains that the recommendation to sway specifically applies to Torah learning or to singing songs and praise. However, in the intense and internal prayer of the Amidah, in which we stand before the King, it is not proper to move at all; only one’s lips may move (Shlah, Masechet Tamid, Ner Mitzvah). Since each minhag has opinions on which to rely, every person may practice in the way that allows him to concentrate the most. This applies especially to a person who became accustomed to shuckling and therefore finds it hard to concentrate while standing still (Magen Avraham, Mishnah Berurah 48:5; and see Kaf Hachaim 48:7-9).