When this Trap Card’s activation resolves, look at the face-up monsters you control. If they all have the same Type/Attribute, you don’t have to do anything else. If there are different Types/Attributes, you choose 1 Type/Attribute to keep, and send your face-up monsters that aren’t that Type/Attribute to the Graveyard.
While Rivalry/Gozen is active and you have a face-up monster, you cannot Summon monsters that are a different Type/Attribute from the monster(s) you already control.
So you can’t do any of the following to put a monster with the wrong Type/Attribute on your side of the field:
Activate a card or effect that can only Special Summon monsters of the wrong Type/Attribute (including effects that could theoretically Summon anything, but currently cannot, such as “Monster Reborn” when you control Spellcasters but the only monsters in the Graveyard are Zombies).
While Rivalry/Gozen is active, you can still Set monsters of any Type/Attribute, and you can Special Summon monsters of the wrong Type/Attribute face-down. But you can’t Flip Summon them afterwards, and they’re sent to the Graveyard after damage calculation if your opponent attacks them. (Also, in that case, you won’t get the effects of cards that need to be destroyed by battle and sent to the Graveyard, because they’re sent to the Graveyard by the Trap Card’s effect instead. But Flip Effects will still activate.)
You can activate cards and effects that would Summon the same Type/Attribute monster as you have on the field.
You can also activate cards that would Summon more than one Type/Attribute of monster. The results depend on what you have on the field:
If you DON’T control any monsters: You can activate cards that would Summon monsters with more than one Type/Attribute. All of them are Summoned, but you must immediately pick one Type/Attribute and send all others to the Graveyard.
If you already DO control a monster: First of all, if the Summoning effect targets, you can only target monsters with the right Type/Attribute.
However, you can also activate cards that would Summon lots of monsters without targeting (like “Return from the Different Dimension”), as long as at least 1 of them is the right Type/Attribute. In this case, only the right monsters are Summoned to the field.
The same rule applies to the “Gladiator Beast” monsters. You can return a Gladiator Beast to your Deck and activate its effect as long as there is another Gladiator Beast in your Deck that you COULD Summon that is the same Type/Attribute as the one you returned. However, if you have no face-up monsters when the effect resolves, you can Summon any Type/Attribute of monster.
You can take control of an opponent’s monster that’s the wrong Type/Attribute with a card like “Mind Control.” If you do, it’s sent to the Graveyard immediately after you gain control of it.
If you control no face-up monsters, you can Summon a monster of any Type/Attribute. Also, if Rivalry/Gozen is being negated, you can Summon monsters of any Type/Attribute. If it stops being negated, you have to send monsters to the Graveyard until you control only 1 Type/Attribute again, just like if Rivalry/Gozen had just resolved.
While "Rivalry of Warlords" is active, when "Creature Swap" is activated, players may exchange monsters of different Types. After a monster switches control, if its new controller also controls a monster of a different Type, then the monster which switched control is sent to the Graveyard. However, if you only control one monster after the monsters switch control, then you only control one Type of monster, so the monster whose control you gained is not sent to the Graveyard.
These TCG rulings were issued by Upper Deck Entertainment and have since been deemed unofficial by Konami. They were previously considered official and can still be correct, as long as they follow the official gameplay rules, or unless Konami has issued a ruling that says otherwise. References: 
When "Rivalry of Warlords" is activated, each player must send monsters to the Graveyard so that he controls only 1 Type of monster face-up. You don't select or name a Type when "Rivalry of Warlords" is activated; instead, you make it so you have only 1 Type. If, after this, your monsters are destroyed so you have no monsters on the field, you are free to play any Type again, and that new Type becomes the Type that you must play. (So if you send monsters to the Graveyard so you control only Spellcaster-Types, and your Spellcaster-Type monsters are destroyed, you may play a Dragon-Type monster, but can then only play Dragon-Type monsters as long as you control 1 or more face-up Dragon-Type monsters. If all your monsters are again destroyed, you can again play whatever Type you want.)
This also means that when "Rivalry of Warlords" is active, each player can control a different face-up Type than his opponent.
You can activate "Rivalry of Warlords" when either or both player(s) does not have any face-up monsters on the field, in which case once a player does have a face-up monster that monster's Type becomes that player's Type for purposes of "Rivalry of Warlords" (as long as the monster remains face-up on that player's side of the field).
When "Rivalry of Warlords" is activated, it has no effect on monsters that are face-down at that time and does not flip those monsters face-up.
If you have 2 monsters on the field face-up of the same Type, you cannot Tribute 1 of them for another monster that is a different Type (even "Jinzo") as you would then have 2 different Types.
If "Cyber Jar" is activated while "Rivalry of Warlords" is active, you can Special Summon different Types of monsters face-down without penalty, or Special Summon 1 Type face-up and all the others face-down. But if you Special Summon different Types of monsters face-up, you have to destroy [sic] monsters to make it 1 Type.[Notes 2]
When "Rivalry of Warlords" is active, and you change the Type of all of your monsters with "DNA Surgery" or "Parasite Paracide" or "D. Tribe", you now only have 1 Type so you are OK and no monsters are destroyed [sic], and you can also Summon, Flip Summon, and Special Summon additional monsters that do not have that printed Type on the card and they will be considered that Type and are not destroyed [sic].[Notes 2]
Sacred Phoenix of Nephthys: If "Rivalry of Warlords" is active, and your opponent takes control of your "Sacred Phoenix of Nephthys" while controlling monsters that are not Winged Beast-Type, "Sacred Phoenix of Nephthys" is not considered destroyed by the effect of "Rivalry of Warlords", but rather by a condition that made it impossible for it to exist on that side of the field. So it is not Special Summoned by its own effect during the Standby Phase.
Uria, Lord of Searing Flames: If you control "Rivalry of Warlords" and 2 other face-up Trap Cards, and you control a non-Pyro monster, you cannot send the 3 face-up Trap Cards to the Graveyard to Summon "Uria, Lord of Searing Flames" because "Rivalry of Warlords" prevents you from Summoning monsters of different Types.
Vampire Lord: When "Vampire Lord" is Summoned while your opponent's "Rivalry of Warlords" is active and your only monsters are non-Zombie-Type monsters, "Vampire Lord" is destroyed [sic], but is not considered destroyed [sic] by "Rivalry of Warlords"' effect and won't be Special Summoned.[Notes 2]
↑Email Rulings are considered unverified, as their source(s) are not publicly viewable. They are not confirmed as applicable to the TCG until they have been officially published by Konami.
↑ 2.02.12.22.32.42.5Monsters sent to the Graveyard by "Rivalry of Warlords" due to their Type are not destroyed.