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We report the discovery of six ultra-faint Milky Way satellites discovered 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 ($M_{V} \geq -3.2$ 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.