Detection of the cosmological neutral hydrogen signal from the Epoch of Reionization, and estimation of its basic physical parameters, is the principal scientific aim of many current low-frequency radio telescopes. Here we describe the Cosmological HI Power Spectrum Estimator (CHIPS), an algorithm developed and implemented with data from the Murchison Widefield Array (MWA), to compute the two-dimensional and spherically-averaged power spectrum of brightness temperature fluctuations. The principal motivations for CHIPS are the application of realistic instrumental and foreground models to form the optimal estimator, thereby maximising the likelihood of unbiased signal estimation, and allowing a full covariant understanding of the outputs. CHIPS employs an inverse-covariance weighting of the data through the maximum likelihood estimator, thereby allowing use of the full parameter space for signal estimation (foreground suppression). We describe the motivation for the algorithm, implementation, application to real and simulated data, and early outputs. Upon application to a set of 3 hours of data, we set a 2$sigma$ upper limit on the EoR dimensionless power at $k=0.05$~h.Mpc$^{-1}$ of $Delta_k^2<7.6times{10^4}$~mK$^2$ in the redshift range $z=[6.2-6.6]$, consistent with previous estimates.
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