How should I deal with "package 'xxx' is not available (for R version x.y.z)" warning?

I tried to install a package, using

install.packages("foobarbaz")

but received the warning

Warning message:package 'foobarbaz' is not available (for R version x.y.z)

Why doesn't R think that the package is available?

See also these questions referring to specific instances of this problem:

My package doesn't work for R 2.15.2
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