Nonparametric statistics is concerned with making inference about aspects of an underlying distribution without enforcing explicit structural assumptions on their form. We study the nonparametric statistical problem of estimating a (Lebesgue) density based on a sample of independent, real-valued random variables. Kernel density estimators are simple and conceptually appealing, and we present finite-sample guarantees on their variance and bias under appropriate smoothness assumptions. These clarify the role of the key tuning parameter associated with the method, namely the bandwidth. We also present two techniques for bandwidth selection, based on least-squares cross-validation and Lepski's method.