so if my url for my website is just random like birthdayparty.com but I sell dog toys (this is not the situation haha), will this affect my ranking in search engines?
I think it does, but you better read the SEO (Search Engine Optimization) forums here: http://forums.digitalpoint.com/forumdisplay.php?f=12 They also have a guide here: http://forums.digitalpoint.com/showthread.php?t=413544
There's a whole bunch of people that will swear it's the magic bullet, but it's not. It does help, but not as much as several years ago. So yes, it does have an effect, but it's arguable as to how much. I've been using exact match domains since 2007 and Google at least has reduced it's importance. But the keyword in the domain does affect it, maybe even more so as in your case where it's unrelated, thus affecting both of the websites/webpages.