How to get column mean for specific rows only?

Since findInterval requires year to be sorted (as it is in your example) I'd be tempted to use cut in case it isn't sorted [proved wrong, thanks @DWin]. For completeness the data.table equivalent (scales for large data) is :

require(data.table)
DT = as.data.table(DF)   # or just start with a data.table in the first place

DT[, mean:=mean(score), by=cut(year,c(-Inf,1984,1991,Inf))]

or findInterval is likely faster as DWin used :

DT[, mean:=mean(score), by=findInterval(year,c(-Inf,1984,1991,Inf))]

datfrm$mean <-
  with (datfrm, ave( score, findInterval(year, c(-Inf, 1984, 1991, Inf)), FUN= mean) )

The title question is a bit different than the real question and would be answered by using logical indexing. If one wanted only the mean for a particular subset say year >= 1984 & year <= 1990 it would be done via:

mn84_90 <- with(datfrm, mean(score[year >= 1984 & year <= 1990]) )