Decomposing a graph into a hierarchical structure via $k$-core analysis is a standard operation in any modern graph-mining toolkit. $k$-core decomposition is a simple and efficient method that allows to analyze a graph beyond its mere degree distribution. More specifically, it is used to identify areas in the graph of increasing centrality and connectedness, and it allows to reveal the structural organization of the graph. Despite the fact that $k$-core analysis relies on vertex degrees, $k$-cores do not satisfy a certain, rather natural, density property. Simply put, the most central $k$-core is not necessarily the densest subgraph. This inconsistency between $k$-cores and graph density provides the basis of our study. We start by defining what it means for a subgraph to be locally-dense, and we show that our definition entails a nested chain decomposition of the graph, similar to the one given by $k$-cores, but in this case the components are arranged in order of increasing density. We show that such a locally-dense decomposition for a graph $G=(V,E)$ can be computed in polynomial time. The running time of the exact decomposition algorithm is $O(|V|^2|E|)$ but is significantly faster in practice. In addition, we develop a linear-time algorithm that provides a factor-2 approximation to the optimal locally-dense decomposition. Furthermore, we show that the $k$-core decomposition is also a factor-2 approximation, however, as demonstrated by our experimental evaluation, in practice $k$-cores have different structure than locally-dense subgraphs, and as predicted by the theory, $k$-cores are not always well-aligned with graph density.