Yaoi is a genre of anime focusing on highly idealized male x male relationships. It's usually written by women for women, as opposed to bara created by 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 or women.
Be careful with the terms- yaoi technically boyxboy sex and yuri is girlxgirl sex. The right terms for romance would be shounen-ai and shoujo-ai respectively. Often, however, yaoi and yuri are used as catch-all terms... which has been known to cause me some confusion. XD
(The picture is NaruSasu... a VERY popular yaoi couple. :3)
Simple, English definition:
Yaoi is love between two guys; "gay guys," if you wish.
Yuri is love between two girls; "lesbians," if that's easier to remember.