A breadboard is a handy way to experiment with electronics with no need for soldering.
On the left and right edges are power rails where all the holes are connected vertically. Connect to a power or ground wire from the Arduino and it's available in all the other holes in that column.
In the middle, the grid of holes is connected in groups of 5 along each row. This allows you to plug in a wire and connect it with up to four other components.
For the breadboard to be useful, you'll also need hookup wires to connect it to the Arduino, and resistors, LEDs, or whatever, to place on the breadboard.
If you don't have a ThinkerShield, you can simulate it with a breadboard, otherwise a breadboard won't be needed for any of the early videos. It will come into its own when we start to cover devices that aren't on the ThinkerShield (e.g. stepper motor).