Monsieur (Sir or Mister, in French), is as common a word in French as it is tricky to pronounce.

In a nutshell: to pronounce it correctly, one needs to forget everything they know about French pronunciation, let alone English pronunciation and visualize an entirely different word.

Made by aggregation of the expression "mon sieur" (my sir, my lord...) it evolved into a word of its own and progressively took a pronunciation of its own too.

You should not pronounce the "r" in Monsieur, neither should you pronounce the "on" the way it is done usually. The correct way is by saying the phonetic equivalent of "meussieu" (in French) or something like "muss-YUH" (probably the closest we can get to the French phoneme "eu") for an English speaker.

You are now able to add an exquisite touch of courtesy when cursing at someone in French or addressing some kind of snappy comeback to a local:
"Monsieur, vos insinuations sur ma prononciation de la langue française me cassent les couilles."
("Sir, you are breaking my nuts with your comments on my pronunciation of the French language").

As for most words in everyday spoken French, "monsieur" is quite often turned into a mumbled "m'sieur". However, this pronunciation would be considered inappropriate in formal social settings. (thanks MarcK)