The pencil-and-paper version of Dungeons & Dragons isn't the same today as it was in 1974. The game's publishers have tweaked the rules extensively since D&D's debut. The most current version of the game uses fourth-edition rules. But the basis for the rules governing Dungeons & Dragons Online is version 3.5.
The game's mechanics rely on statistics and chance. Characters have attributes like strength, dexterity, wisdom and intelligence. You quantify each attribute with a number. In the traditional paper game of D&D, players do this by rolling dice and assigning the outcomes to attributes. In general, these attributes can range from three (very poor) to 18 (superior). Other factors such as the character's race or equipment can influence these attributes.
When a player wishes his or her character to complete a task, the game master must take into consideration both the character's skills and the difficulty of the task. The player then rolls a die (or several dice) to insert an element of chance in the attempt. If the die roll falls within the range of success, the character completes the task. In some cases, such as if the character is searching for a secret door, the game master might roll the die behind a screen so that the players can't see the result. The game master explains what happens next -- the character fails to find a door. The players don't know for sure whether they couldn't find the secret door, or whether there was ever a secret door to begin with.
Dungeons & Dragons Online follows the same model. As your character attempts to perform tasks such as hitting a monster with a weapon or disarming a trap, the game randomly generates a number and compares that to the difficulty of the task. With some tasks, you can make multiple attempts to succeed. Other tasks have a skill level or attribute threshold your character must meet to complete them. If your character's abilities fall below that threshold, you won't succeed no matter how many times you try.
Turbine didn't create a slavishly-accurate computer representation of the pencil-and-paper version of the game. Throughout the development process, the programmers and game architects decided that some aspects of the paper version of the game wouldn't translate well in an MMORPG setting. The most obvious departure is in how the game handles combat.