Glad to see a new player!
During combat, creatures deal damage to each other simultaneously. If at any point a creature takes damage equal to their toughness, they die. Damage can come from combat, spells, or abilities, and likewise toughness can also be reduced through spells or abilities.
There are a number of keyword abilities which modify these rules; I'll only touch on a few:
Lifelink: When this creature deals damage for any reason, its controller gains that much life. It doesn't matter how much toughness a blocking creature has; an 8/8 lifelink will still gain its controller 8 life even if it's blocked by a 1/1 creature.
First Strike: When this creature blocks or is blocked by another creature(s), it gets to deal its damage first. If multiple creatures in a blocking situation have first strike, they deal damage simultaneously, then all other creatures deal damage.
Double Strike: Nasty ability. This creature gets to deal damage as if it has first strike, then it deals damage again when everybody else gets to attack.
Trample: If this creature would deal lethal damage to blocking creatures and still have damage left over, the leftover damage is dealt to the opponent instead of being wasted.
Deathtouch: A single point of damage dealt by this creature is lethal to a creature. It is not lethal to you.
So in your case, you had a 4/3 creature with first strike and lifelink. He had a 1/3 deathtouch creature. When they fight, your creature deals 4 damage first. This kills the scorpion before it has a chance to fight back. You gain 4 life.