In Arkham Horror, each Investigator has six skills. Encounter cards will often demand that you make a specific skill check to accomplish a task. A couple of these skills are used for two things. Confusingly, the original rule layout didn't do such a good job of highlighting the difference between these two things. I'll try to do a better job.
At the start of the game, you may position your skill sliders wherever you wish. These choices may be adjusted later during the upkeep phase. This is governed by the Investigator's focus (see below). Note that when one skill increases, another reduces. For example, if you use the slider to increase your Speed skill, your Sneak skill gets lower because of it.
Fight is the skill that is used for physical skill checks. For example, a card might say "A hooded figure grabs you. Make a Fight check to wrestle him to the ground." In this case, you'll make a Fight check, modified by any skills, allies, etc that specify an increase to Fight.
Fight is also the base skill that governs Combat checks. A Combat check is a check made specifically during a combat with monsters, not a skill check demanded by a card. Anything that modifies the Fight skill also modifies the Combat check, but the inverse is not true. For example, if you have the common item card .45 Automatic, you gain a +4 modifier to Combat checks, but this modifier doesn't apply if a Fight check is what's called for.
Lore represents knowledge more than anything else. For example, the description for reading a magical tome might say "The words move about the page. Make a Lore check to read them."
Lore is also the base skill that governs Spell checks, to determine whether your spell is successful or not.
Luck represents simple chance. An encounter card might say "You pass the night at the roadhouse playing cards. Pass a Luck check to gain $5." Luck checks are the most frequent checks called for in this manner.
Sneak is the skill that is used when a check calls for you to use stealth to get by. For example, an encounter card might say "You happen on a group of robed figures chanting next to the fire. Pass a Sneak check to sneak past them."
Sneak is also the base skill that governs Evade checks. Evade checks are made not at the demand of an encounter card, but when you're specifically trying to remain hidden from a monster. Like Fight vs Combat checks, anything that modifies Sneak modifies Evade checks, but not the other way around.
Speed is the skill check called for when you need to do something in a hurry. An example card might say "The boulder rolls down the hill, threatening to block off your only way into the cave. Make a Sneak check to get there first."
Speed also determines how far an Investigator may move during their turn.
Will is a measurement of the Investigator's force of willpower and personality. An encounter card might say "The miasma fogs your brain, threatening to take you over. Pass a Will check to assert yourself."
Will is also the base skill that governs Horror checks. A Horror check is made at the sight of a monster so alien and horrific that it threatens to break the Investigator's brain just to see it. Like Fight vs Combat checks, anything that modifies Will modifies Horror checks, but not the other way around.
During any skill check, the number of the skill plus any modifiers (positive or negative) equals the number of dice the player rolls during the check. For example, If an Investigator's Sneak skill is currently set at 3 and they have a skill card grating +1 to Sneak checks and they have an item granting +2 to Sneak checks and there is a weather card in effect that reduces sneak checks by 1, then when a Sneak check is called for, the player will roll 5 dice (3+1+2-1=5).
In most skill checks, only one success is required to pass the check. Occasionally, a skill check will require multiple successes. This is most often the case in Combat checks. Note, also, that skill checks will often give their own modifiers to the check. For example, a card might demand that you pass "a Sneak -1 check." In this case, the modifier is included with any other modifiers the Investigator has.
Here's the easy part. All dice rolled in Arkham Horror are d6. By default, any roll of a 5 or a 6 is a success. Thus, if you need 2 successes on a skill check and you roll 6 dice for it and get a result of 1,3,3,5,6,2, then you have passed the check. If an Investigator becomes Blessed, then rolls of 4, 5 or 6 are counted as successes. If an Investigator becomes Cursed, only a 6 is counted as a success.
In addition to being used for sealing Gates, Clue tokens can be used to assist with skill checks. By spending a Clue token, a player may roll another d6. This can be done even after seeing the result of the original roll. And it may be done over and over again, so long as there are Clue tokens to spend.
As they are such large parts of Arkham Horror, Combat checks Horror checks and Evade checks deserve a how-to. Typically, an Investigator will encounter a monster either because he moves into an area with a monster in it, a monster moves into the Investigator's area or the player draws an Encounter card that says "A hideous monster appears!" In the case of an Investigator and a monster sharing the same area, there usually only 2 options, fight or flee.
Sometimes, an Investigator won't want to fight a monster. Maybe he doesn't have a decent weapon. Maybe he's in a hurry. Maybe he knows it'll use him to wipe its ass. The way to get past a monster without having to fight is to succeed on an Evade check. And Evade check uses the Investigator's Sneak, plus and modifiers to Sneak checks and Evade checks. It is also modified by the number in red on the upper right corner of the monster tile.
Using the Mi-go above as an example, we can see that it's pretty perceptive and it modifies Evade checks by -2. So if our Investigator has a Sneak skill of 3, plus a skill that adds 1 to sneak, plus a magic item that adds 1 to sneak, then the player will roll 3 dice. Only one success is required for an Evasion check. If there are multiple monsters in the same area as the Investigator, the Investigator must succeed on checks against each monster.
If an Evade check is made as part of movement, the Investigator may keep moving. It's assumed he snuck past the monster. If anything that halts movement happened prior to the Evade check (for example, if the Investigator initiated Combat against one monster, won, then decided to Evade a second monster), then the Investigator may not leave the area. It's assumed he found a hiding spot.
If an Evade check fails, the monster immediately deals its combat damage to the Investigator. This number is represented by the number of red hearts in the lower right corner of the monster's tile and is applied to the Investigator's stamina. If the Investigator is still conscious after this damage, he must make a Horror check (see below). If he is still sane following the Horror check, he may attempt to fight or flee again. Combat continues until an Evade check is passed, the monster is dead or the Investigator is unconscious or insane.
The first thing that must happen when an Investigator decides to fight a monster is the Horror check. This represents the cumulative damage to the Investigator's sanity that comes from seeing things that mortal man was never meant to lay eyes on. A Horror check uses the Investigator's Will, modified by anything that modifies Will checks and Horror checks. It's further modified by the blue number in the bottom left corner of the monster's tile. If this number is instead a simple blue line, then no Horror check is required. The number of blue dots below the number is the sanity damage the Investigator will take if he or she fails the Horror check.
Using the Mi-go above as an example, we can see that the number in blue is -1. If the Investigator has a Will of 2 and no other modifiers, the player will roll 1 die. Only one success is required for a Horror check.
No matter how long combat takes, the Investigator only has to roll the Horror check one time per monster per combat. Whether he passes or fails, he still only has to roll the once.
With the Horror check done, a Combat check is next. A Combat check uses the Investigator's Fight, plus any modifiers to Fight checks and Combat checks. It's also modified by the red number in the bottom right corner of the monster's tile.
Using the Mi-go above as an example, we see this number is +0. if the Investigator has a Fight skill of 4, plus an ally card that increases Fight by 1, plus a weapon that adds 4 to Combat checks, then he or she rolls 9 dice. The target number of successes is equal to the number of blood drops in the bottom center of the monster's tile. In the Mi-go's case, this is only one. If the monster has multiple blood drops, then that number of successes must be rolled all at once, not over successive Combat checks.
If the Combat check fails, the monster deals its combat damage to the Investigator's stamina. If the Combat check succeeds, the Investigator has killed the monster and may (usually) take it as a spendable monster trophy.
Take note that many monsters have special abilities printed on their monster tile. Some are no big deal. Some can hurt you even if you kill them. It's important to know exactly what you're fighting before you commit yourself to combat.
Spellcasting deserves its own brief run through as well. Magic usually comes with a price. Its use can sometimes drive people mad. But it's powerful. Some spells are simple and act the same as weapons. But some are very powerful. That monster that's blocking off half the map, has an enormous Evade modifier and is immune to both physical weapons and magical weapons? Well, there are spells that can banish it. Hell, there are spells that can take it out of the game completely so it can't randomly come back later.
To cast a spell the investigator must first pay the sanity cost, if any, listed in the spell. This is taken directly as damage to the Investigator's sanity. Using the Wrack spell above, this is 1, so the Investigator would take a single point of sanity damage. If the damage does not drive the Investigator insane, then next an Investigator must make a Spell check. A Spell check uses the Investigator's Lore skill, modified by anything that modifies Lore checks or Spell checks. Most spells have a casting modifier, which modifies an Investigator's Spell check. In the case of the Wrack spell above, it's -2. So if an Investigator has a Lore of 4 and no other modifiers, the player will roll 2 dice. Only one success is needed to pass a Spell check. This done, the effect of the spell becomes available to the Investigator.