It's relatively easy to make a row of LEDs flash in sequence using simple digital electronics. Don't be scared by the word "digital": this project doesn't require any programming, just a pair of common logic chips that cost pennies. You'll use a chip called an "inverter" to create a simple oscillator, and use the oscillator to drive a chip called a "counter" that will light the LEDs successively. As with anything in electronics, there are other ways to accomplish this, but this is a fairly simple solution that uses a minimum of parts.
Insert both ICs into the breadboard. A breadboard is a gadget that makes it easy to build prototype circuits. It's a grid of pinholes over a copper base. All the pinholes in each column are connected electrically, and the columns are isolated from each other. At each end of the breadboard there are two rows of pinholes that are separate from the columns. Common practice is to use one of the rows for the positive supply voltage, and the other row for ground. That way, everything that needs to connect to power or ground can just be connected to the appropriate row. A line down the centre of the breadboard divides it into two sections. Connect the ICs to the breadboard by fitting the centre line of each IC over the centre line of the breadboard. Taking care not to bend any pins, push the chips firmly into the breadboard so that the pins fit into the pinholes. This way, each pin on each chip has its own column of pinholes.
Connect the supply voltage and ground pins on the inverter chip to the power supply and ground rows on the breadboard, respectively. Consult the inverter's data sheet for its pin diagram. The diagram will show you which pins are for power and ground, and which pins are the input and output for each inverter section.
Turn one of the inverter sections into an oscillator by connecting the input to ground through a capacitor and then connecting the output to the input through a resistor. The inverter IC actually includes several inverters, each with its own input and output pin. An inverter takes a digital input and flips it: if the input is low, the output is high, and vice versa. The resistor and capacitor work together to keep the inverter flipping back and forth, creating an oscillator. Connect a component to a pin on the IC by plugging one of the component's leads into a pinhole on the breadboard on the same column as the pin.
Make the correct power and ground connections for the counter chip. Again, consult the data sheet for the pin diagram and description.
Connect the output from the inverter oscillator to the input of the counter.
Connect each output of the counter to the anode of an LED. When the counter fires a pulse out of an output, the LED connected to that output will light up.
Calculate the resistor value you need for your LEDs using Ohm's law. Look in your LED's data sheet for its forward voltage and current. Subtract the LED's forward voltage from the circuit supply voltage to get the voltage drop across the resistor. If you're using a nine volt battery, the supply voltage is nine volts. Divide that by the LED current (in amps, not milliamps) to get the minimum resistor value in ohms. Resistors come in standard values, so use the next highest value you have available. If you don't have the LED's documentation available, 2 Volts and 0.02 Amps (20 ma) is a good rule of thumb. If you're in doubt, any resistor between about 400 and 1000 ohms should do the trick. Using lower values will make the LEDs glow more brightly, but if the resistance is too low the LEDs will burn out.
Connect the cathode of each LED to ground through a resistor with the value you calculated in the previous step.
Connect the battery clip to the breadboard. The battery clip should have two wires sticking out of it, red and black. Red is positive voltage and black is ground. Connect the red wire to the row on the breadboard you want to use for the supply voltage, and connect the black wire to the row you are using for ground. Plug the battery into the clip, and watch the LEDs blink.
Data sheets for any chip ("integrated circuit," or "IC") are available free online from the manufacturer and vendors. A data sheet provides all the relevant details you need to use an IC. You need to use a special type of inverter called a "Schmitt Trigger Inverter." A Schmitt trigger is a sort of filter that smoothes out the action of the inverter. The simple resistor-capacitor oscillator used in this project will only work with a Schmitt trigger inverter, not a regular inverter. Schmitt trigger inverters are widely available in ICs that package several inverter sections on a single chip. The value of the resistor and capacitor in the oscillator control its frequency. The higher the values, the slower the oscillator. Experiment with different combinations of components to find a speed you like. You can make the speed variable by replacing the resistor with a potentiometer. A potentiometer is a variable resistor with a knob you can turn to control it. It has three terminals, and turning the knob changes the resistance between the centre terminal and the two on the outside. Connect the centre tap of the potentiometer to the inverter input, and connect the right-hand terminal to the inverter output. Counter ICs cycle through their outputs every time there's a pulse at the input. For this project, a decade counter may be the most useful. A decade counter has ten outputs. Every time it receives a pulse at the input (from our oscillator) it sends out a pulse from a successive output, and resets to the first output every ten input pulses.
Electrolytic capacitors are polarised. If you use an electrolytic capacitor, the cathode (marked with a "-" symbol on the casing) must connect to ground. Reverse-biasing an electrolytic capacitor will damage it.