The n-solvable filtration of the link concordance group, defined by Cochran, Orr, and Teichner in 2003, is a tool for studying smooth knot and link concordance that yields important results in low-dimensional topology. We focus on the first two stages of the n-solvable filtration, which are the class of 0-solvable links and the class of 0.5-solvable links. We introduce a new equivalence relation on links called 0-solve equivalence and establish both an algebraic and a geometric characterization 0-solve equivalent links. As a result, we completely characterize 0-solvable links and we give a classification of links up to 0-solve equivalence. We relate 0-solvable links to known results about links bounding gropes and Whitney towers in the 4-ball. We then establish a sufficient condition for a link to be 0.5-solvable and show that 0.5-solvable links must have vanishing Sato-Levine invariants.