The Wizard gritted his teeth and watched the oncoming wave of demons pour out of the cave’s mouth, knuckles white as he gripped the smooth staff in his hand. Dragging it along the floor in a wide arc, he kicked the end of it up over his head in a flourish. With the wrath of a raging storm, the tip of the staff slapped the ground and shot out a bolt of lightning, tearing through the horde and leaving behind nothing but the smell of burnt meat.
An Arcane Focus is a tool that allows certain classes to cast a spell with a material component without needing to provide that component. An example of this is casting Lightning Bolt, using a staff as a focus instead of providing rabbit fur and a glass rod, like the Wizard in the introduction did.
There are two exceptions: if a spell says it consumes the material component or requires a material component with a listed gold piece cost, then said component must be provided and cannot be replaced by a focus.
The classes which can use an Arcane Focus are the Sorcerer, Warlock, and Wizard. These classes can choose one of the following as their Focus:
A specially constructed staff
A wand-like length of wood
Your DM may allow you to use other appropriate items as a Focus. If you are playing in the world of Eberron, you may be able to gain benefits from different types of Foci, such as minor extra fire damage if your wand is made of a particular wood.