When binary black holes form in the field, it is expected that their orbits typically circularize before coalescence. In galactic nuclei and globular clusters, binary black holes can form dynamically. Recent results suggest that ≈5% of mergers in globular clusters result from three-body interactions. These three-body interactions are expected to induce significant orbital eccentricity 0.1 when they enter the Advanced LIGO and Virgo band at a gravitational-wave frequency of 10 Hz. Measurements of binary black hole eccentricity therefore provide a means for determining whether or not dynamic formation is the primary channel for producing binary black hole mergers. We present a framework for performing Bayesian parameter estimation on gravitational-wave observations of eccentric black hole inspirals. Using this framework, and employing the nonspinning, inspiral-only EccentricFD waveform approximant, we determine the minimum detectable eccentricity for an event with masses and distance similar to GW150914. At design sensitivity, we find that the current generation of advanced observatories will be sensitive to orbital eccentricities of 0.05 at a gravitational-wave frequency of 10 Hz, demonstrating that existing detectors can use eccentricity to distinguish between circular field binaries and globular cluster triples. We compare this result to eccentricity distributions predicted to result from three black hole binary formation channels, showing that measurements of eccentricity could be used to infer the population properties of binary black holes.