We present Hubble Space Telescope (HST ) Space Telescope Imaging Spectrograph observations of two extreme narrow-line Seyfert 1 galaxies ( NLS1's), IRAS 13224À3809 and 1H 0707À495. The spectra are characterized by very blue continua; broad, strongly blueshifted high-ionization lines (including C iv and N v); and narrow, symmetric intermediate-(including C iii], Si iii], and Al iii) and low-ionization (e.g., Mg ii) lines centered at the rest wavelength. The emission-line profiles suggest that the high-ionization lines are produced in a wind and that the intermediate-and low-ionization lines are produced in low-velocity gas associated with the accretion disk or base of the wind. In this paper, we present the analysis of the spectra from these two objects; in a companion paper, we present photoionization analysis and a toy dynamical model for the wind. The highly asymmetric profile of C iv suggests that it is dominated by emission from the wind, so we develop a template for the wind from the C iv line. We model the bright emission lines in the spectra using a combination of this template and a narrow symmetric line centered at the rest wavelength. We also analyzed a comparison sample of Hubble Space Telescope spectra from 14 additional NLS1's and constructed a correlation matrix of emission-line and continuum properties. A number of strong correlations were observed, including several involving the asymmetry of the C iv line.