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