ARIMA is one of the classic forecasting models, combining 3 ideas into one.
AR uses past values, I stands for integrated and means differencing, and MA uses past forecast errors.
The combination handles trends and many kinds of short-term pattern.
The order is written ARIMA(p, d, q), with p past values, d differences and q past errors.
For example, ARIMA(0, 1, 0) is a random walk, forecasting tomorrow as today.
ARIMA(0, 1, 1) is closely related to simple exponential smoothing.
The ACF and PACF of the differenced series help choose p and q.
Automatic methods try many orders and pick the best by an information criterion, a score rewarding fit and penalising complexity.
Seasonal ARIMA, written with a second set of orders, handles patterns that repeat each week or year.
Extra columns such as holidays can be added, a variant often called ARIMAX.
Prediction intervals come from the model and widen as the horizon grows.
ARIMA assumes a stable relationship over time, so sudden breaks hurt it.
Very long seasonal periods, such as 365 days in daily data, make seasonal ARIMA slow and unstable.
In practice ARIMA is a strong baseline, and newer methods should beat it on past data before replacing it.
The usual mistake is confusing the MA part, which uses past errors, with a moving average of the values.