What exactly is the exponential smoothing model? I see the term "exponential smoothing" model used a lot in different applications but I never understood what exactly it is.  Is it just a MA(1) model?  Or is it any moving average model, meaning it can be any MA model?  
For example, the Airline Model has a specific order, which is ARIMA(0,1,1)(0,1,1).  Does the exponential smoothing model have a specific order or characteristics that make it an exponential smoothing model?  What makes an ARMA model an exponential smoothing model?
 A: Strictly speaking the name Moving Average is misleading, since it is a moving average of errors, not of values themselves. You can see this by comparing the expressions for MA(1) and Simple Exponential Smoothing: 
Simple Exponential Smoothing

$\hat{Y}_t = \alpha Y_t  + (1-\alpha )\hat{Y}_{t-1} $    

with $Y_t$ the actual at time t, and $\hat{Y}_t$ the forecast at time t. 
MA(1):

$\hat{Y}_t = \mu + \epsilon_t + \theta\epsilon_{t-1}$

with $\hat{Y}_t$ the forecast at time t, and $\epsilon_t = \hat{Y}_t - Y_t$ the error at time t. 
Because of this, SES is actually equivalent to ARIMA(0,1,1) not MA(1) with ($\theta =\alpha-1$).
To answer the title question: That is all exponential smoothing is. 
The other more complex exponential smoothing methods, such as double and triple exponential smoothing are variations on this that add trend, seasonality and dampening. As such, exponential smoothing models don't have orders or degrees of complexity the way ARIMA models do.
For the relationship between ARMA/ARIMA and the more general family of exponential smoothing, see here. 
