Never having played any of the original D&G games, I find all of these abreviated rolls of the dice very confusing and would appreciate one of you experts out there in never, never land taking a moment to explain it to me using the following example.. Given a)Longspear+4
To hit +4/damage +4
vs
b)Greataxe+3
To hit +3/damage +3
And the question is: which one does the most damage and how do you calculate it? And in the "to hit" line, are the repeated set of numbers ie +4 and +4 redundant or are they both significant? Appreciate your help...
Dungeons & Dragons 1D8+4vs1D12+3
Now - I've not played this game myself, but have played several D&D games 
Firstly:
The "1D8" - this means 1 roll with a dice that can result in numbers 1 to 8 (8 sided dice).
Likewise 2D4 would be 2 rolls with a 4 sided dice which will result in a number between 1 and 4. (with a total result ranging between 2 and 8)
A roll with a standard 6 sided dice would thus be mentioned as 1D6.
The "To Hit" number is added to your "To hit" roll, to test if you actually hit with the weapon.
The damage number is added to your damage when you have hit the enemy.
So first - you roll an attack roll with a 20 sided dice, yielding a result between 1 to 20.
Then your to-hit modificers are added, wether they be from your character statistics (strenght for meele weapons) or the weapon (bonus "to hit", for instance the +3 and +4).
The total of your attack roll is then compared with the Armour Class of your opponent. If you hit, another roll is made to calculate the damage of your blow.
As for which weapon does most damage of the two you listed:
1D8+4 equals damage ranging from 5 to 12 whereas your 1D12+3 gives 4 to 15. (There will likely be added damage to both of these from your strenght modifier.)
So you shouldn't only think of damage per hit, but also how often you will hit. It dosen't matter you yeild the most powerfull weapon in the world, if you can't hit your enemy. But you also should take into account if you wish a higher base damage or you wish to do the potentially most damage.
Such desecions are often very subjective - but personally - I prefere to go with highest possible "to-hit" and highest possible base damage.
Hope this helps a little.
Firstly:
The "1D8" - this means 1 roll with a dice that can result in numbers 1 to 8 (8 sided dice).
Likewise 2D4 would be 2 rolls with a 4 sided dice which will result in a number between 1 and 4. (with a total result ranging between 2 and 8)
A roll with a standard 6 sided dice would thus be mentioned as 1D6.
The "To Hit" number is added to your "To hit" roll, to test if you actually hit with the weapon.
The damage number is added to your damage when you have hit the enemy.
So first - you roll an attack roll with a 20 sided dice, yielding a result between 1 to 20.
Then your to-hit modificers are added, wether they be from your character statistics (strenght for meele weapons) or the weapon (bonus "to hit", for instance the +3 and +4).
The total of your attack roll is then compared with the Armour Class of your opponent. If you hit, another roll is made to calculate the damage of your blow.
As for which weapon does most damage of the two you listed:
1D8+4 equals damage ranging from 5 to 12 whereas your 1D12+3 gives 4 to 15. (There will likely be added damage to both of these from your strenght modifier.)
So you shouldn't only think of damage per hit, but also how often you will hit. It dosen't matter you yeild the most powerfull weapon in the world, if you can't hit your enemy. But you also should take into account if you wish a higher base damage or you wish to do the potentially most damage.
Such desecions are often very subjective - but personally - I prefere to go with highest possible "to-hit" and highest possible base damage.
Hope this helps a little.
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Morpheus33
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I'll give it a go
two fighters, one with an AC of 23, the other who is the attacker has +9 to hit
so rolling a 20 sided die to attack(which the computer does for you), the attacker must roll 14 or higher to hit the man with the AC of 23( roll of 14 + attack bonus of +9= 23)
so the higher the attack bonus the better, the higher the AC the better
two fighters, one with an AC of 23, the other who is the attacker has +9 to hit
so rolling a 20 sided die to attack(which the computer does for you), the attacker must roll 14 or higher to hit the man with the AC of 23( roll of 14 + attack bonus of +9= 23)
so the higher the attack bonus the better, the higher the AC the better