I am looking for a database that would provide classification of various types of RNA - information about its structure, function, etc.
Rfam is a database about RNA families, similar to Pfam (wich is for proteins). I think it's what you're looking for. http://rfam.sanger.ac.uk/search
You could also enjoy miRbase, a database for micro RNAs: http://www.mirbase.org/ There are tRNA databases, too, like this one: http://trnadb.bioinf.uni-leipzig.de/ And this tool for predict RNA secondary structure (However, this kind of tools aren't as good as they proteinic counterpart): http://mfold.rna.albany.edu/?q=mfold/RNA-Folding-Form