To initiate Player vs. Player combat, you have to attack another player. You may do so as your Action on your turn - just move into another player's space and announce you are attacking him.
Exceptions: you cannot attack a player that is in a Magic portal or a city (this is not applicable for the first game, since players won't be entering cities). Multiple figures can occupy these spaces without fighting.
Also, you may not attack any player after the end of a Round has been announced - in this case, you can't end your movement on another player's space (see End of the Turn - Forced Withdrawal if this happens accidentally).
Tactical note: When you have no cards in your hand (and your Units are spent), you are a pretty easy target for a Player vs. Player attack. If the end of the Round has been announced, you are safe, and you can easily spend all your cards. But imagine this scenario - you have five cards in your hand, but no cards in your Deed deck. Now, you have to decide - if you announce the end of Round, you cannot use these last five cards. But if you don't and play them all, you will be an easy target until your next turn, when you will have a chance to announce the end of Round again. Watch your opponents and consider their plans carefully (especially when they are close to you on the map).
Reaction Of The Defender
The attacked player (we call him the Defender) announces whether he wants to fully attend the attack (that means he takes his full turn in advance, and uses all benefits that would provide), or whether he just considers it to be an annoying disturbance (and thus does not sacrifice his entire turn to fight the opponent).
Read the appropriate part of Rulebook for more details.
Tactical note: The second option is usually used if the attacked player is sure the attack will be not very serious and does not want to be diverted from his plan. It can also be used if he thinks the aggressive player is stronger and that he intends to keep attacking. By holding on to some Move cards, he can try to get away from the aggressor during his next turn. If he attends the attack fully, he uses up his turn to do so, and he will have no chance to escape or do anything else before it is aggressor's turn again.
There are only two phases of Player vs. Player combat.
Ranged and Siege Attacks phase - players take turns performing Ranged and Siege Attacks, starting with the Defender. These attacks can be used to Wound the enemy Hero or his Units. The phase ends when no player wants to use such attacks.
Melee Attacks phase - players take turns performing any Attacks, starting with the Aggressor. Contrary to the Ranged and Siege Attacks phase, these Attacks do not necessarily have to be used to assign damage - players may also use them to force the opponent out of the space, or to steal his Artifacts. This phase (and the entire combat) ends when one player is forced out of the space, or if no player wants to continue to use attacks (in which case, the Aggressor has to withdraw).
See the appropriate part of Rulebook for more details.
Differences Between Player vs. Enemy and Player vs. Player Combat To help you understand Player vs. Player combat better, we summarize the two main differences here:
When fighting enemies on the map, you either block the entire attack, or it goes through at full strength; there is nothing in between. In Player vs. Player combat however, you are allowed to partially block an attack; this reduces the amount of damage it would deal.
Note: Blocks performed in the Ranged and Siege Attacks phase are halved (round down). It is difficult to block these attacks - this way, they are similar to the attacks of enemies with the Swift ability.
When fighting enemies on the map, the player assigns the damage from their attacks. That means, he decides whether and which Units will be hit, and how much damage goes to his Hero. On the other hand, the damage is rounded to his disadvantage, i.e. even one point of damage can Wound a Unit with high Armor.
In Player vs. Player combat, the attacking player gets to assign the damage he deals. However, damage is rounded to the advantage of the defending player - the attacker has to have enough damage if he wants to Wound a Unit or player (or do any of the other allowed effects), and it may happen that some damage remains unused (as there is not enough for the attacker to do anything with it).