The Field is the area where Duelists place their Cards. Playmats are often used as the Field.
The Field is divided into two halves; one for each player, with each half having five Main Monster Zones, five Spell & Trap Zones (including two Pendulum Zones), a Field Zone, a Deck Zone, an Extra Deck Zone, and a Graveyard. In the middle of the Field, there are two Extra Monster Zones, which do not belong to either player while unoccupied.
Referenced by Card Effect
The "Field" referenced by card text is the combination of each player's Monster Zones, Spell, & Trap Zones, Pendulum Zones, and Field Zones. Thus, the Field is shared and each player controls only one side of it.
The Field is also unofficially referred to as “the board”.