We present results from an archival study of 70 medium-redshift QSOs observed with the Wide Field Planetary Camera 2 on board the Hubble Space Telescope. The QSOs have magnitudes M_V <= -23 (total nuclear plus host light) and redshifts $0.06 <= z <= 0.46$. The aim of the present study is to investigate the connections between the nuclear and host properties of QSOs, using high-resolution images and removing the central point source to reveal the host structure. We confirm that more luminous QSO nuclei are found in more luminous host galaxies. Using central black hole masses from the literature, we find that nuclear luminosity also generally increases with black hole mass, but it is not tightly correlated. Nuclear luminosities range from 2.3% to 200% of the Eddington limit. Those in elliptical hosts cover the range fairly evenly, while those in spirals are clustered near the Eddington limit. Using a principal components analysis, we find a kind of fundamental plane relating the nuclear luminosity to the size and effective surface magnitude of the bulge. Using optical nuclear luminosity, this relationship explains 96.1% of the variance in the overall sample, while another version of the relationship uses x-ray nuclear luminosity and explains 95.2% of the variance. The form of this QSO fundamental plane shows similarities to the well-studied fundamental plane of elliptical galaxies, and we examine the possible relationship between them as well as the difficulties involved in establishing this connection. Comment: 49 pages, 6 figures. Accepted by ApJ