what are the five different classes of RNA molecules found in eukaryotes?

The three that come straight to mind are messenger RNA (mRNA), transfer (rRNA), and ribosomal RNA (rRNA).  There are also small nuclear RNA (snRNA).
In short - messenger RNA are the transcripts of parts of DNA (genes) that encode proteins.  Transfer RNAs carry the amino acids that are assembled together in the ribosome.  Ribsomal RNA is found only in ribosomes and is required for the operation of the ribosomes.  Ribosomes are like chloroplasts in that they have their own DNA.  Small nuclear RNAs encode a range of small nuclear ribonucleic proteins that help splice introns out of a raw transcripts.

There is also microRNA (miRNA) but I'm not sure what this is