A random error is an experimental error (or in standard English, a source of measurement errors in an experiment - not just an error in one specific result) which falls evenly on both sides of the true results; typically the measurements will be distributed in a Gaussian curve around the true value of the thing being measured. This will reduce the accuracy the results and increase their apparent range, but multiple runs will bring the average result ever-closer to the true value being measured. Random error is sometimes thought of as noise.
Contrast this with the systematic errors caused by things like incorrectly calibrated instruments, which consistently skew results in a particular direction and therefore cause mistakes in measurements which persist however many times an experiment is repeated.
Every experiment will include random errors to a greater or lesser extent, which is the main reason almost all experiments include multiple runs. Classic sources of random error include inaccuracy in the reading of measuring instruments or the recording of results, and minor, arbitrary variations in the experimental setup. Sometimes it is possible to eliminate or greatly reduce the random error from particular sources, through careful control of the conditions of the experiment; other times the best approach is simply to repeat the experiment many times.
Mostly remembered from physics lectures; http://trochim.human.cornell.edu/kb/measerr.htm was the single best web page about experimental errors that I could find.