Run Charts are sometimes used when there is insufficient data to properly analyze a process using statistical control charts. They should never be used in place of control charts, and cannot offer the benefits of control charts (mainly that of ascertaining whether a process is in a state of statistical control, or to determine process capability).
The x-axes are time based, so that the charts show a history of the process. For this reason, you must have data that is time-ordered; that is, entered in the sequence from which it was generated.
Run Charts typically use three of the standard eight Western Electric / Nelson Run Tests Rules. These three rules provide a broad indication of process stability, but by themselves lack the sensitivity of statistical control charts.