As above, anyone willing to take out the mathematical jargon and notations - i can get that from any book on time series and explain what really is happening, why and how? Surely, there is someone who understands that these methodologies were developed to solve real world problems and not complicate things as they generally do for math-illiterates like me.
Here's a stab:
Single Exponential Smoothing: We use a "level component", which you can interpret as the underlying "reality" of the time series. It is initialized as the first value of the series (or in a more sophisticated way).
After that, it is updated with every new observation, as a weighted average between the new observation and the previous level component.
After the history, we forecast out using the current level.
Double Exponential Smoothing: As with single smoothing, we use and update a level component. In addition, we use a "trend component". It is initialized, e.g., as the trend between the first two observations (or between the first and the last observation, or something else). With each observation, we update both the level and the trend component, in each case as a weighted average of the previous component and an appropriate value derived from the new observation.
Of course, the update step becomes more tricky, since we need to take the level component into account when updating the trend component, and vice versa.
For the forecast, we take the last level component and extrapolate the last trend component out.
Triple Exponential Smoothing: As with double smoothing, but we also use "seasonal components": one for each seasonal bucket (so, 12 components for monthly data). We initialize these, e.g., as the difference between the first year's observations and the initial level and trend components. With each observation, we update the level, the trend and the appropriate seasonal component. For instance, in January, we update the seasonal component relevant to January.
The update step is of course even trickier than with double smoothing, since all components hang together and need to be taken into account in updating.
The forecast then uses the current level component, an extrapolated trend component and the appropriate seasonal component.
All this can be done with additive or multiplicative trends, seasons and errors. And trends can (and should) be dampened. For more information, look at Gardner's Exponential Smoothing - the state of the art part II in the International Journal of Forecasting (2006), or at Forecasting with Exponential Smoothing - the State Space Approach by Hyndman et al.