These are what your girl friends will become if they think you're going after their guy (which you most likely are, right?) They're not that tough to fight, if you follow some general guidelines. :)
Who likes Who
Be grateful only 4 guys have girls other than you chasing after them, so you won't have to go through the torture of rivalry every game. :\ Below is the list of those four and your corresponding rival.
Saukya - Arisawa
Kazuma - Konno
Madoka - Natsumi
Mihara - Sudou
From Friend to Rival
This happens when your girl friend is in a lower state with you than the guy she likes. E.g. she's smiling but he's already grinning. Each of the girls have different ways of going on rival mode, which can be found in their individual section in the girls page. After they have declared their rivalry (some more subtly than others *cough*sudou*cough* o_0) when you check the stats page with your bro you'll find this colourful 'VS' icon decoration next to their name.
There are four potential ways to fight your rival, the former 2 to prevent her from becoming your rival and the latter 2 to stop her from winning your guy.
Do I know you?
The first and the easiest (I can't reiterate this enough) is to never ever befriend her. Meaning? Do not choose the room associated with her (or you'll be forced to play with her to up your social stats) and after you meet her (which you will once your relationship with her 'guy' improves) don't play with her. And she'll never bother you. :)
Make sure she's always one state higher or the same as your guy. That means, if he's smiling, she's gotta be grinning, and so on. You don't have to worry if both of them reach tokimeki state with you, by default you'll get the guy's ending.
I hate you!
Once she goes on rival mode, stop playing with her. I've never actually done this before (the first is so much simpler ^^) but it works for other people so I'm assuming it'll be okay.
I love you!
Once she goes on rival mode, play with her like crazy and butter her up till she loves you so much she gives her guy up to you. How considerate. ^_~ I tried this once, but it failed (though then I was actually aiming for Himuro-sensei). Of course, I might not have played with her enough.
From Rival to Friend
There are two ways this will happen.
The first and the worst is that she has snagged your guy. She'll come prancing up to you with the dude and practically flirt with him in front of you (she'll call him by his pet name, grr).
The second and much better one is that she tells you she's giving up on the guy. I haven't had this one before so what I know I picked up from Klist.
Once she's friendly with you again, the 'VS' sign will disappear from her name in your bro's stats page and you'll get her emails, cards, visits etc once more.