Rivals[]
You will face players around your level and also dependent on your alliance size. In addition, you can only attack players 30 levels below yours, but can attack any other player above your level. A level 75 player can attack level 45-480 players. However, this limit is removed if the player was given a bounty or attacked by a player 30 levels lower then them; if a level 4 player attacks a level 35, the level 35 can attack the level 4 player despite a gap of 31 levels between them.
To win the battle, you must have ATK around or higher then the opposing player's DEF. The higher your ATK is, the greater your chances are of winning. If you want to have the best chances of winning against a tough opponent, you should set a Dark monster as your leader as their leader effect provides ATK bonus.
When you win a battle, you will gain [(1-5)*2] experience, losing some health in the process. You will also gain gold if the opponent had gold in their coffers. The gold taken is dependent the opponent's level; not by your level. If you lose, you will only gain [(1)*2] experience.
As such, other players can also attack you. If they win, you will lose health and take a percentage of your gold if you had any on you. If you need to defend yourself better, set a Holy monster as your leader as their leader effect provides DEF bonus.
Regardless of attacks, you lose health anywhere from 1-9 (wins) and 16-24 (losses). More monsters will mean more damage dealt/health lost, so try to balance a good amount of monsters when you can. If you lose 8-9 health fairly frequently when attacking someone, you should not attack them as you will have a greater chance of losing compared to losing 1-4 health.
Once a player falls below 26 health, they cannot be attacked until they recover back up to the said amount.
Before you attack a player, you should check what monsters they have. This is longer then using binoculars, which cost 
2 and can scan an opponent's attack and defense values, including their skill level in those stats. By checking a player's monster size, you can determine if it is worth to attack the said player or not.
Bounty[]
You can place other players on the bounty board, and so can other players do so to you.
To start a bounty, select a player's profile, then tap on the "Bounty" option. Before you can put them on the bounty board, you must assign a value of gold which is given to the player who KO's the selected target. The value of gold can be increased to 100x the normal amount. Bounty can be given to any player of any level, regardless of who created the bounty. This means a level 420 player can bounty a level 4 player, and vice versa.
The gold increases as the player levels up, and if the player is lower then a higher level player. For example, we will assume the bounty cost for a level 4 player is 500 gold, and a level 420 player is 953,000. If the level 4 player tries to bounty a player lower then or equal to their level, it will cost 500 gold. If the same player tries to bounty a level 420 player, it will cost them 953,000 gold.
When a player is on a bounty list, it follows the similar rules as Rival battling. However, any player can attacked the bountied target, even if they are more then 30 levels greater then them. To obtain the bounty, the target must be KO'ed; in this case they must reach 0 health for it to happen. Bounties are related to KO Wins and Losses which are relatively hard achievements to obtain as usually, many strong players fight over most bounties while lower levels are reluctant to attack stronger opponents.
Bounty is often used for PB (punchbags), as they serve as damage sponges for online players, giving free experience to them without having to attack others, especially for higher levels, where their potential targets are more likely to be active and have strong monsters. You can expect a PB to have massive amounts of health to tank damage from all players.
Alliance Revenge[]
In the Alliance Revenge section, you can set Alliance Revenge on anyone, similar to bounty. However, the person is an exclusive target for your Alliances made with other friends, and follows the same rules as Rival Battle, excluding the level limit. It costs 10 jewels to revenge someone, and lasts for 24 hours as soon as they are put on revenge.
Whenever your alliances attack the target and win, you will be notified in the news when it occurs. Alliance Revenge should not be used unless the target has been attacking you frequently, as giving them a bounty instead is cheaper since the jewels can be used for other things, such as monster eggs and boss attacks.
Boss Battle[]
Boss Battle has 3 different sections. In battling, with the succession of defeating the boss, players will receive rewards based on your contribution.
Dungeon[]
You can find dungeons randomly when completing quests. You can join dungeons from other players by selecting Dungeon on Home tab. You get summoning bonus in form of a random legendary monster, when you find AND enter the dungeon. Dungeon rewards are monsters only. More information here.
Team Battle[]
Team Battles are 6v6 fights against members of other teams. Monsters are selected are based on your current strongest monster of that type. More information here.
Arena[]
In the arena, you choose your team of max 6 monsters and fight other people's teams for rewards.
Tower of Masters[]
Your selection of monsters will fight waves of enemy monsters. All monsters have to be from the same tier or lower. More information here.