  The detection of ~50 coalescing compact binaries with the Advanced LIGO and
Virgo detectors has allowed us to test general relativity, constrain merger
rates, and look for evidence of tidal effects, compact object spins, higher
waveform modes, and black hole ringdowns. An effect that has not yet been
confidently detected is binary eccentricity, which might be present in a small
fraction of binaries formed dynamically. Here we discuss general limits on
eccentricity that can, in-principle, be placed on all types of compact object
binaries by a detector operating at the design sensitivity of Advanced LIGO.
Using a post-Newtonian model for gravitational-wave phasing valid in the small
eccentricity regime, we assess the relative measurement error for eccentricity
for a variety of spinning and non-spinning binaries. Errors and correlations
involving the mass and spin parameters are also investigated. We find that
decreasing the low frequency limit of a detector's observational frequency band
is one of the key design factors for increasing the odds of measuring binary
eccentricity. We also introduce and analytically explore the eccentric chirp
mass parameter, which replaces the chirp mass as the key measurable parameter
combination in eccentric gravitational waveform models. The eccentric chirp
mass parameter explains a degeneracy between the chirp mass and the
eccentricity. This degeneracy leads to a bias in the standard chirp mass
parameter. We also investigate the systematic parameter bias that arises when
eccentric systems are recovered using circular waveform templates. We use both
Fisher matrix and Bayesian-inference-based Markov Chain Monte Carlo (MCMC)
methods to investigate these parameter estimation issues, and we find good
agreement between the two approaches (for both statistical and systematic
errors) in the appropriate signal-to-noise ratio regime. (abridged)
