As an example imagine a 60 W light bulb running off 120 V. The current will be 1/2 A. Now assume that you put a second 60 W bulb in series with the first. Now the resistance of the single 60 W bulb is 240 Ohms. The 1/2 A flowing through the bulb heats up the filament wire and causes an amount of light to be emitted. Resistance in series adds, so the total resistance for two of the bulbs in series is 480 Ohms and by Ohms law V = I x R, the current through each bulb will be 1/4 amp and hence the bulbs will be less bright.
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