Gemara

(Beraisa): There are three things that should not pass between two people, and one should not pass between a pair of them - dogs, date trees and (men sandwiching or being sandwiched by) women;

Some say, also pigs; some say, also snakes.

Rav Papa: If one passed between them, he should say the verses that begin and end with 'Kel' or 'Lo'.

Rashi says that the verses are "Kel Motzi'am mi'Mitzrayim..." and "Lo Ish Kel..."; R. Chananel says that they are "Kel Hash-m Hatizleni..." and "Lo Titsheni..."

If a Nidah walked between two men:

If she was beginning Nidah, this will cause one of them to die (some say that she can kill one through witchcraft);

If she was at the end of Nidah, this will cause the men to quarrel.

Rav Papa: If she passed between them, they should begin and end with 'Kel'.

Horayos 13b: Ten things cause difficulty in learning...passing between two women, or a woman passing between two men.

Maharsha (Horayos 13b DH veha'Over): Saying the verses protects from harm, but does not fix the difficulty in learning.

Rishonim

Tosfos (DH Niftach): We may not use Divrei Torah to heal, but we may use them to protect.

Poskim

Kitzur Shulchan Aruch 3:8: A man should not pass between two women, dogs, or pigs; two men should not allow any of them to pass between them.

Ben Ish Chai (2:Pinchas 17): One may not walk between two women. It is permitted if they are more than four Amos apart, or if two men walk together between them.

However, the Ben Ish Chai (Ben Yehoyada Pesachim 111) says if a Nidah walked between two men, one of them will be damaged only if a Mazik was present at the time. Since nowadays Mazikim are not frequent, we need not be careful about this.

The Lev Eliyahu was careful not to walk between girls (Lev Eliyahu Bereishis Ma'asim v'Hanhagos p. 26). He cites Rav Simchah Zisel of Kelm that we follow hidden matters like this exactly as stated; 'Nashim' excludes virgins. When there is no alternative, one may rely on a stick or handkerchief to be a Chatzitzah between himself and women.

Sefer Zikaron (2:19, printed in Si'ach ha'Sadeh): It is forbidden only if at least one of the three is stationary (Teshuvos Salmas Chaim 2:35). Minors are not called 'Nashim' (Yevamos 61b).

Kol Eliyahu (the Vilna Gaon, on Esther 5:4): [According to the opinion that Esther became Nidah – Megilah 15a], she invited Haman to the party in order to be between him and Achashverosh. This will cause one of the men to die (if at the beginning of Nidah), or (if at the end of Nidah), cause them to quarrel. Rav Elyashiv (on our Daf) - we may infer that Ru'ach Ra'ah applies to Goyim. This is relevant to a Goy who slept over food.