This is a common question. In base, the option you're looking for is aggregate. Assuming your data.frame is called "mydf", you can use the following.
> aggregate(B ~ A, mydf, sum)
A B
1 1 5
2 2 3
3 3 11
I would also recommend looking into the "data.table" package.
> library(data.table)
> DT <- data.table(mydf)
> DT[, sum(B), by = A]
A V1
1: 1 5
2: 2 3
3: 3 11
