HoMA/The Core Mechanic
The core mechanic for this game is simple. Roll a d20 and add a stat + other modifiers (there aren't many of these). Higher is better.
Combat
In combat most of the times you will need to roll is when it is your turn and you are making an attack, there are two attacks everyone can make, the ranged and melee basics. To make an attack you roll a d20 and add the listed bonus on your character sheet. If it hits you will then deal however much damage the attack specifies, as well as do any other things that the attack may list.
You will most likely:
Roll against physical defense using your Strength or Dexterity and your weapon to make a melee or archery based attack
Roll against mental defense using your Wisdom or Intellect and your symbol to make a magic based or compulsion type attack
Non Combat
Out of combat most checks you will make are to use one of your skills, areas you specialize in, to accomplish something that has a chance of failure. Skills that are for your class are always rolled adding 5 + your level, skills that you do not specialize in, you add a relevant ability score to do.
You will most likely:
Roll against a Target Difficulty using a skill to accomplish heroic tasks
Roll against a Target Difficulty using a Stat to accomplish tasks outside your specialty