Is there a public database in this world consisting of known number fields, their discriminants, and their ideal class groups, etc? If so, how does a lay person like me have access to this database?
Hideo Wada, A table of ideal class groups of imaginary quadratic fields, Proc. Japan Acad. Volume 46, Number 5 (1970), 401-403 is available at http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pja/1195520300&page=record.
I found that by typing $$\rm table\ class\ group$$ into Google, and I expect more can be found by that technique.
From PARI's tables: This directory contains a stripped down version of the number field tables published by the Bordeaux computational number theory group (H. Cohen, F. Diaz y Diaz, M. Olivier and their students) around 1995, and incorporates a number of corrections. The original [uncorrected] tables can be found at ftp://megrez.math.u-bordeaux.fr/pub/numberfields/
AUTHORS: See ftp://megrez.math.u-bordeaux.fr/pub/numberfields/readme.pdf for the authors of the original tables (~1995). The present version was set up by Karim Belabas (Bordeaux) for the PARI group (2007).
FORMAT: The file T.gp contains data pertaining to fields of degree $\rm\:3 \le n \le 7\:$ with $\rm\:0 \le r \le n\:$ real places. The data is in GP format, one field per line, meant to be fed to the gp calculator as in
v = readvec("T31.gp");
which stores in the vector $\rm\: v\ (182417$ elements) data corresponding to the $182417$ complex cubic fields with discriminant $\:\!> -10^6.$ Reading the largest table requires a PARI stack size around $\rm\:\!30\:\!M$. Entries in the resulting vector are sorted by increasing discriminant absolute value. Each entry is a $4$-components vector: [disc, V, h, cyc], where