We have used the Planetary Camera on the Hubble Space Telescope to study the morphology and surface brightness parameters of a luminosity- limited sample of fourteen elliptical galaxies in the Virgo cluster. The total apparent blue magnitudes of the galaxies range between 9.4 and 13.4. In this paper, the core brightness profiles are presented, while the overall morphology and the isophotal shapes are discussed in two companion papers [Jaffe et al. AJ, 108, 1567 (1994); van den Bosch et al. AJ, 108, 1579 (1994)]. We show that, in spite of the spherical aberration affecting the HST primary mirror, deconvolution techniques allow recovery of the brightness profile up to 0.2arcsec from the center of the galaxies. We find that none of the galaxies has an isothermal core. On the basis of their morphological and photometrical properties, the galaxies can be divided in two physically distinct groups, referred to as Type I and Type II. All of the Type I galaxies are classified as E1 to E3 in the Revised Shapley Ames Catalog (Sandage & Tammann 1981), while Type II galaxies are classified as E5 to E7. The characteristics of Type II galaxies are explained by the presence of disks component on both the 1 arcsec and the 10 arcsec scales, while Type I galaxies correspond to the classical disk-free ellipticals. Addresses: Ferrarese L. ferrares@jhufos.pha.jhu.edu Ford H.C. ford@jhufos.pha.jhu.edu Johns Hopkins University and Space Telescope Science Institute, Baltimore, Maryland 21218 van den Bosch F.C. vdbosch@strw.leidenuniv.nl Leiden Observatory, Leiden, The Netherlands Jaffe W. jaffe@strw.leidenuniv.nl Leiden Observatory, Leiden, The Netherlands O'Connell R.W. rwo@perseus.astro.virginia.edu University of Virginia, Charlottesville, Virginia 22903