As with many inventions there is no simple answer, most inventions are just refining what someone else has already done.
Some monitors use a cathode ray tube, same as most TV's, etc. There are other options such as plasma and TFT.
Much info about CRT on this site:
http://inventors.about.com/library/inventors/blcathoderaytub e.htm
Answer to your question:
1855 - German, Heinrich Geissler invents the Geissler tube, created using his mercury pump this was the first good evacuated (of air) vacuum tube later modified by Sir William Crookes.
1859 - German mathematician and physicist, Julius Plucker experiments with invisible cathode rays. Cathode rays were first identified by Julius Plucker.
1878 - Englishmen, Sir William Crookes was the first person to confirm the existence of cathode rays by displaying them, with his invention of the Crookes tube, a crude prototype for all future cathode ray tubes.
1897 - German, Karl Ferdinand Braun invents the CRT oscilloscope - the Braun Tube was the forerunner of today's television and radar tubes.
1929 - Vladimir Kosma Zworykin invented a cathode ray tube called the kinescope - for use with a primitive television system.
1931 - Allen B. Du Mont made the first commercially practical and durable CRT for television.