From Unofficial BOINC Wiki
A projection forwards in time. In Climateprediction.net (CPDN) it referrs to where a climate model is run for a future period of time with estimates of future Forcings. It is often run after a Hindcast which allows researchers to see how well that model performs. The Transient Coupled Model includes both a Hindcast and a Forecast.
The Forecast is what is wanted but the Hindcast is needed to assess how good the Forecast is likely to be.