We present ground-based optical observations of GRB 020124 starting 1.6 hr after the burst, as well as subsequent Very Large Array and Hubble Space Telescope (HST) observations. The optical afterglow of GRB 020124 is one of the faintest afterglows detected to date, and it exhibits a relatively rapid decay, F$_ ν$åisebox-0.5ex t$^-1.60+/-0.04$, followed by further steepening. In addition, a weak radio source was found coincident with the optical afterglow. The HST observations reveal that a positionally coincident host galaxy must be the faintest host to date, R>i̊sebox-0.5ex 29.5 mag. The afterglow observations can be explained by several models requiring little or no extinction within the host galaxy, A$^host$$_V$rs̊ebox-0.5ex 0-0.9 mag. These observations have significant implications for the interpretation of the so-called dark bursts (bursts for which no optical afterglow is detected), which are usually attributed to dust extinction within the host galaxy. The faintness and relatively rapid decay of the afterglow of GRB 020124, combined with the low inferred extinction, indicate that some dark bursts are intrinsically dim and not dust obscured. Thus, the diversity in the underlying properties of optical afterglows must be observationally determined before substantive inferences can be drawn from the statistics of dark bursts.