Breadboards are plastic boards, covered with holes, that are used for prototyping electronics projects. Instead of soldering components together, you can just stick the components and some wires into the board. The components and wires are easy to insert, easy to change and easy to remove. Underneath the boards, the holes are connected electrically in patterns of rows and columns.
Make a Radio
The classic electronic project is building a radio. When you first hear the voices and music coming out of a tangle of components and wires, you have passed a threshold in project building. Breadboards make this easy because you do not have to solder the components together. Just insert them into the board. Usually you connect the power supply along one side of the board and the ground along the other side. Plug in the components on the schematic of the radio circuit to the rows of holes in between. You can easily put together a working radio in an hour or less.
Make a Disco Style Light Panel
It takes a little more creative designing to build a disco style light panel. You can start with trying to activate a patch of LEDs on one corner or your breadboard, but as the project progresses you will probably want to get a second breadboard and cover it with LEDs. The patterns of LEDs turning on and off can be controlled several different ways: by sound as they are in discos by putting a microphone in the circuit and separating the different frequencies with different values of capacitors, by touch or by displaying preprogrammed patterns. This is a project where you are only limited by your imagination.
Build Computer Circuits
Depending on your knowledge of computer circuits, there are several good breadboard projects. An easy project is a decoder, like decoding keystrokes into ASCII strings, or decoding binary numbers into decimals. A slightly harder project is encoding and decoding Hamming or other error correcting codes that are used in data transmission. The classic computer circuit to build is an adder -- a circuit that adds two binary strings. This is really a digital design project and you can build the adder using logic gates on chips. If you are more adventurous, you can build your own logic gates out of transistors and resistors.
Many robot kits have tiny breadboards built in. You can connect components to sensors and motors -- and sometimes other components -- to control the behavior of the robot. You can also connect from this breadboard to the microprocessor that is programmable through your computer. This chip also makes an excellent controller for the disco light panel project. You can connect the programmable chip directly into your breadboard to control other projects.