Cross Processing is intentionally processing film in the wrong chemicals, creating interesting and unpredictable color shifts and increased contrast. So for example, cross processing would be shooting a roll of color slide film or E6 and developing it as if it were color negative film or C41 (or visa versa).
Tip: Cross Processing is also commonly referred to as Xpro
Positive color slide film cross processing
E6 => C41: taking positive color slide film (E6) and processing it in color negative chemicals (C41). This is the most common type of cross processing. This typically creates increased contrast with strong color casts.
Color negative film cross processing
C41 => E6: taking color negative film (C41) and developing in slide chemicals (E6). You can expect muted pastel colors with little contrast. You may want to increase contrast by asking the lab to push 2 or 3 stops.