A Direct Attack is the process where a Monster's whole ATK value is subtracted from the opponent's Life Points, either when there are no Monsters on your opponent's side of the field, or if the attacking monster has the ability to attack directly, either by its own effect or the effect of another card. The damage inflicted as a result is known as "Direct Damage". There is almost always a risk in a direct attack as the opponent frequently has a spell or trap card set as a counter.
Many monsters have the ability to perform a Direct Attack, even when there is a monster on the opponent's side of the field. However, to balance this, they typically have very low ATK values or an added disadvantageous effect (for example, "Crystal Beast Amethyst Cat" has an effect that allows you to halve that card's ATK to allow it to attack your opponent directly). Other ways to attack directly includes special cards that allow this effect (Ex. "Opti-Camouflage Armor", "Battle Teleportation"). Usually it's after you attack a monster and destroy it successfully.
One major exception to the rule of monsters with direct attack capabilities are the "Earthbound Immortals", powerful monsters that cannot be attacked themselves. They are instead balanced by destroying themselves automatically if there is no Field Spell Card on the field.
It should be noted with great importance that inflicting Direct Damage is a primary win condition for almost all decks.