Says person to the right rolls a Might 4 attack. What does that mean? Do they roll 4 dice? Also, after they roll and I roll, what damage do I take, if any? There are other similar cards, as well.
It means that you get attacked following all the normal rules of an attack from a person or monster with a might of 4. The person to your right is simply taking the role of rolling the dice.
So he will roll 4 dice, and you will then roll dice equal to your might. If his total is greater than yours, you will take physical damage equal to the difference of the rolls. If you tie or win, nothing happens.
rolls a Might 4 attack
These are Betrayal specific terms and they have a very distinct meaning. Attacking is done with the Might trait (unless stated otherwise) and it says the player rolls a Might attack at strength of 4. Thus the player would roll four dice as if they had a Might level of 4.
what damage do I take, if any
Damage in Betrayal is always a difference of pips. If Alice attacks Bob (Traitor) and Alice rolls 3 Might dice and gets 5 pips, and Bob rolls to defend and rolls 5 Might dice and gets 8 pips. Then the difference is 3 pips and the lower number takes the damage. In this example Alice would take 3 damage (even though she was the attacker) to her physical traits (Might, Speed).
If Bob had rolled 2 pips then Bob would take 3 damage to his physical traits.
If Bob had rolled 5 pips then since number is the same nothing happens.
Keep in mind that ordinarily if you cause someone 2 pips of damage instead of dealing damage you have the option to steal from them. In this case it's a card attacking you not another explorer so it doesn't apply but it's still something of which you might wish to be aware.
Relevant Quotes: Page 8
When you make an attack, roll a number of dice equal to your Might. Your opponent does the same. Whoever rolls a higher result inflicts physical damage to the other explorer or monster. That player defeats his or her opponent. The amount of damage equals the difference between the two rolls. [..] If there's a tie, no one gets hurt.
You can't use a trait to attack an opponent who doesn't have that trait. For instance, if a monster doesn't have Sanity, you can't make a Sanity attack against it.
Stealing Items: If you attack someone and inflict 2 or more points of physical damage, you can steal a tradable item instead of doing the damage. You can't do this with a distance attack.