You roll as many d6s as your Attribute + Skill for the intended action. Each dice rolling a 5 or 6 is a "hit". If there is a difficulty, the "net hits" are the amount on hits above that difficulty.
A player has to roll Charisma + Etiquette, with a difficulty of 3. His
character has 5 in Charisma and 3 in Etiquette. He rolls 8 d6s, and 4
of them roll a 5 or 6. That's 4 hits, or 1 net hit.
"Basic" combat mechanic
1 - The attacker rolls the appropriate attack skill + attribute + modifiers.
A street samurai fires a single shot at a non-moving target. He rolls Agility + Pistols, for 4 hits.
2 - The defender then rolls the appropriate defense skill + attribute + modifiers.
This depends a LOT on how the target defends. For our example, let's say the target just rolls his reaction to jump on the side, for 1
3 - You then compare their results. If the defender rolls more hits than the attacker, or on a tie, the attack misses. If the attacker rolls more hits, the attack succeeds and the net hits is added to the weapon's Damage Value.
If the target had rolled 4 hits or more, nothing would have happened. The attacker rolled more though, so the 3 net hits are added
to the weapon's Damage Value. The weapon had a Damage Value of 6P, it
becomes 9P for this attack.
4 - If the attack succeeds, the defenser then rolls Body + Armor - Weapon Penetration. Each hit reduces Damage Value + net hits of the attack by 1.
The target has a Body + Armor of 7. The gun has a Penetration of -1. The target therefore rolls 6 dice, with each hit reducing the
modified Damage Value by 1. He gets 2 hits, reducing the modified
Damage Value to 7P.
5 - Finally, you apply the final Damage Value to the defenser as wounds (i.e. nothing if it is reduced to 0).
The defenser takes 7 wounds, and crosses 7 boxes of damage on his sheet.
This is the combat mechanic reduced to its "simplest". However, keep in mind that a LOT of modifiers and exception then apply, depending on whether the attacker or defender is moving (or both!), whether anyone is behind cover, the weapons rate of fire, who is already wounded, recoil, and so on.
Drones and Magic
Drones and Magic are both two extremely complex and detailed systems of the game, therefore it is almost impossible to sum them up in a couple of sentences. I will give some guidelines, but take them with a truckload of salt.
Drones have their own mechanic, but basically and in a lot of cases, there will be a hacker using his Mental attributes + hacking skills to attack/defend instead of standard attribute/skills.
As for Magic, there are even more things to take into account. In a lot of situations, spells will be cast using a Mental Attribute and the Spellcasting skill, and will be resisted either through Willpower or normal defending attributes. They also cause damage to the casters. Remember though that there are a LOT of different spells, of ways to cast them, as well as Adepts and Spirits.