19XX is a placeholder year. Alternate versions include 19??, 19xx, 19—, 19__, etc. 199X is also sometimes used. However, since the 21st century, many have found this to be outdated.
The use of 19xx is common in fiction to indicate that the events in the work will take place in the near future, or took place during a specific century. By doing this, the author tries to increase the sense of realism.
The convention is also used on demonstration memos and other paperwork instead of a real date, or, as "19__", for later entry of the actual year. This can be seen on bank checks.
For literature set in the past, any century is substitutable for "19XX", such as Treasure Island's "17__" or Dr. Jekyll and Mr. Hyde's "18--". For current use "20XX" has replaced 19XX in the new millennium.