Edit: This writeup refers to the Japanese word yuri. For the word used by English-speakers, Frater 219's definition is more accurate.
Yuri literally means "lily" in Japanese. When taken individually, the two Chinese characters are hyaku (hundred) and au (to meet), respectively. I'm not sure why those characters came to have this meaning when used together, but this is used as a symbolic element in the short story Yumejûya by Natsume Sôseki. Jim Breen's Japanese-English Dictionary lists 55 names all pronounced "Yuri," of which only one has the same characters (and thus meaning) of the word above. In addition, there are other names such as "Yuriko," Yurika," and "Sayuri" that are sometimes based on the word. Many of these names are exclusively feminine, but others may be unisex or surnames.
The term yuri is commonly used to describe novels, comics, and other media involving relationships (usually, but not always, romanticand, in my experience, not sexual most of the time) between two women. This is not really the gender counterpart to yaoi, but rather the BL (boys' love) genre. Judging by the names of authors, yuri comics and novels are written by both women and men.
The apparent majority of specifically yuri-themed stories are set in high schools in modern Japan, often all-girls' and Catholic schools. In many cases, the two heroines take on sisterly roles (with the younger calling her partner "o-Nee-sama"), or are simply sempai and kôhai. In other cases, one of the two will have a more masculine appearance, mannerisms, or speech patterns. Those where the two are on equal footing tend to begin with either being childhood friends or one transferring into the other's class from another school.
MEDIA WITH YURI ELEMENTS THAT ARE COMPARATIVELY WELL KNOWN IN ENGLISH-SPEAKING COUNTRIES
(I can only assume that those are the target audience for E2. . . .)
『カードキャプターさくら』(CardCaptor Sakuracomics, animation, etc.): Although it was toned down a bit down in the animated version (apparently to appeal to a wider audience), there is an overtone of Daidôji Tomoyo's strong feelings for the title character.
『少女革命ウテナ』(Shôjo Kakumei Utena/La Fillette Revolutionaire/Revolutionary Girl Utenacomics, animation, etc.): Aside from the title character's especially affectionate best friend, there is constant ambiguity about the relationship between Utena and her "bride". At least one female-female relationship also appears among supporting characters, but if I said more, that would spoil the ending of that episode. . . .
『美少女戦士セーラームーン』(Bishôjo Senshi Sailor Moon/Pretty Soldier Sailor Mooncomics, animation, etc.): Aside from the usual camaraderie arising from the premise of an all-female sentai, the supporting characters Ten'ô Haruka (Sailor Uranus) and Kaiô Michiru (Sailor Neptune) are a popular couple to dôjinshi artists.
I am not aware if any of these are commercially available in English-speaking countries, but they can be ordered from amazon.co.jp or jpqueen.com.
『～ｅｓ～エターナルシスターズ』(~es~ Eternal Sisterscomics): A variety of yuri-themed stories.
『百合姉妹』 (Yuri Shimai/Lily Sistersmagazine): In addition to comics and short stories, this newly created quarterly magazine has articles discussing the various yuri-themed media on the market, as well as one columnist's life experiences in girls' love.
『百合天国』(Yuri Tengoku/Girls Heavencomics): Anthologies of yuri-themed stories in school settings.
ONE LAST PLUG
『マリア様がみてる』(Maria-sama ga Mite'ru/La Vierge Marie Vous Regarde/Maria-sama is Watchingnovels, comics, animation): A greatly popular series in Japan, to the point that Yuri Shimai didn't feel necessary to give any description in its section on the series. It centers around the student council of a Catholic girls' school, where upperclassmen adopt underclassmen as their "little sisters" under the soeur system.
See Also: Shojo ai