We present a high-resolution elemental-abundance analysis for a sample of 23 very metal-poor ([Fe/H] <‑2.0) stars, 12 of which are extremely metal-poor ([Fe/H] <‑3.0), and 4 of which are ultra-metal-poor ([Fe/H] <‑4.0). These stars were targeted to explore differences in the abundance ratios for elements that constrain the possible astrophysical sites of element production, including Li, C, N, O, the α-elements, the iron-peak elements, and a number of neutron-capture elements. This sample substantially increases the number of known carbon-enhanced metal-poor (CEMP) and nitrogen-enhanced metal-poor (NEMP) stars—our program stars include eight that are considered “normal” metal-poor stars, six CEMP-no stars, five CEMP-s stars, two CEMP-r stars, and two CEMP-r/s stars. One of the CEMP-r stars and one of the CEMP-r/s stars are possible NEMP stars. We detect lithium for three of the six CEMP-no stars, all of which are Li depleted with respect to the Spite plateau. The majority of the CEMP stars have [C/N] > 0. The stars with [C/N] < 0 suggest a larger degree of mixing; the few CEMP-no stars that exhibit this signature are only found at [Fe/H] <‑3.4, a metallicity below which we also find the CEMP-no stars with large enhancements in Na, Mg, and Al. We confirm the existence of two plateaus in the absolute carbon abundances of CEMP stars, as suggested by Spite et al. We also present evidence for a “floor” in the absolute Ba abundances of CEMP-no stars at A(Ba) ∼ ‑2.0. Based on observations made with the European Southern Observatory telescopes.