A player's connected empire is the territories connected by border, that is owned solely by that player.
A group of territories that when owned by one player, entitle that player to a region bonus
A piece of the gameboard that a player can control. In standard Risk, each territory is represented by a country, or part of a country.
The largest army on the board. The Artillery (or Cannon) represents 10 armies.
The middle army available on the board. Each Cavalry (lighthorse) represents 5 armies.
The standard army used in Risk. Each Infantry (or foot soldier) represents 1 army. They are the most used piece in Risk.
Dice are used to battle in Risk
See: Risk Cards
Risk Cards are traded in for extra troops. In some editions of Risk (example Star Wars Risk), Risk cards play a greater role.