Different Time series Modeling techniques?
ARIMA is a parameterized model that consists of a few parts. First the I is for integrated, and it has to do with a technique to remove non-stationarity from the data, by looking at differences between sequential values instead of the values itself, the parameter indicates how often we want to do this. AR stands for auto-regressive, it's a regression model that regresses on previous values of itself (lag features), the order determines how far back we look. The MA means Moving Average, which is referring to a linear combination of error terms, this is also parameterized by an order. 

Box-Jenkins is a method that uses ARIMA, but it first involves a number of checks before fitting the model, and afterwards does criticism to see how well the model works. This allows for iteration into a better fitting model.

Holt-Winters methods are used for smoothing out time series. Sometimes there is a lot of noise involved in time series, while you are only interested in trends and how values relate to each other. By smoothing it out via for example moving averages or exponential smoothing hopefully you gain more information by trying to filter out noise than you lose by throwing away relevant information.