We report observations of Faraday rotation measures for a sample of 191 extragalactic radio jets observed within the MOJAVE program. Multifrequency Very Long Baseline Array observations were carried out over 12 epochs in 2006 at four frequencies between 8 and 15GHz. We detect parsec-scale Faraday rotation measures in 149 sources and find the quasars to have larger rotation measures on average than BL Lac objects. The median core rotation measures are significantly higher than in the jet components. This is especially true for quasars where we detect a significant negative correlation between the magnitude of the rotation measure and the de-projected distance from the core. We perform detailed simulations of the observational errors of total intensity, polarization, and Faraday rotation, and concentrate on the errors of transverse Faraday rotation measure gradients in unresolved jets. Our simulations show that the finite image restoring beam size has a significant effect on the observed rotation measure gradients, and spurious gradients can occur due to noise in the data if the jet is less than two beams wide in polarization. We detect significant transverse rotation measure gradients in four sources (0923+392, 1226+023, 2230+114, and 2251+158). In 1226+023 the rotation measure is for the first time seen to change sign from positive to negative over the transverse cuts, which supports the presence of a helical magnetic field in the jet. In this source we also detect variations in the jet rotation measure over a timescale of three months, which are difficult to explain with external Faraday screens and suggest internal Faraday rotation. By comparing fractional polarization changes in jet components between the four frequency bands to depolarization models, we find that an external purely random Faraday screen viewed through only a few lines of sight can explain most of our polarization observations, but in some sources, such as 1226+023 and 2251+158, internal Faraday rotation is needed.