やおい is a genre of アニメ focusing on highly idealized male x male relationships. It's usually written によって women for women, as opposed to bara created によって and for gay men. (The men in bara are much more... manly. ^^) Yuri features girl x girl relationships and can be targeted at either men または women.
Be careful with the terms- やおい technically boyxboy sex and yuri is girlxgirl sex. The right terms for romance would be shounen-ai and shoujo-ai respectively. Often, however, やおい and yuri are used as catch-all terms... which has been known to cause me some confusion. XD
(The picture is NaruSasu... a VERY 人気 やおい couple. :3)