We report the discovery of six ultra-faint Milky Way satellites identified through matched-filter searches conducted using Dark Energy Camera (DECam) data processed as part of the second data release of the DECam Local Volume Exploration (DELVE) survey. Leveraging deep Gemini/GMOS-N imaging (for four candidates) as well as follow-up DECam imaging (for two candidates), we characterize the morphologies and stellar populations of these systems. We find that these candidates all share faint absolute magnitudes (M V ≥ −3.2 mag) and old, metal-poor stellar populations (τ > 10 Gyr, [Fe/H] < −1.4 dex). Three of these systems are more extended (r 1/2 > 15 pc), while the other three are compact (r 1/2 < 10 pc). From these properties, we infer that the former three systems (Boötes V, Leo Minor I, and Virgo II) are consistent with ultra-faint dwarf galaxy classifications, whereas the latter three (DELVE 3, DELVE 4, and DELVE 5) are likely ultra-faint star clusters. Using data from the Gaia satellite, we confidently measure the proper motion of Boötes V, Leo Minor I, and DELVE 4, and tentatively detect a proper-motion signal from DELVE 3 and DELVE 5; no signal is detected for Virgo II. We use these measurements to explore possible associations between the newly discovered systems and the Sagittarius dwarf spheroidal, the Magellanic Clouds, and the Vast Polar Structure, finding several plausible associations. Our results offer a preview of the numerous ultra-faint stellar systems that will soon be discovered by the Vera C. Rubin Observatory and highlight the challenges of classifying the faintest stellar systems.