This is a subject that has been covered by many authors. I'll point out Jonathan Kirsch's God Against The Gods as an especially readable account.

First and foremost, if there was one thing the Romans respected, it was ancient and venerable customs and rituals, and they were quick to respect the ancient status of Judaism (a status Christianity lacked later on). The Romans were puzzled by the Jews (the famous if-likely-apocryphal story of Pompey storming into the Temple demanding to see this God of the Jews with his own eyes is a good example), and found the Jews troublesome, puzzling, and annoyingly independent-minded, but until the First Roman-Jewish War kicked off in the 60s CE they avoided direct conflict.

Even later, they were willing to make the occasional compromise with the Jews on account of the venerability of their tradition. Jews praying to their God to bestow favor upon the Emperor was usually an accepted substitute to praying to the Emperor, for example.

Flavius Josephus' The Antiquities Of The Jews is an interesting read if you want to see a Romanized Jew try to give a plot summary of the Old Testament to a pagan readership who did not grow up with those now-famous Biblical stories.

They were a little confused -- Romans and Greeks generally practiced what's called Interpretatio Romana, where they looked at the new gods they were encountering and said "Lugus is just your name for Apollo" or "You call this god Ammon, we call him Zeus". This isn't exactly possible with monotheism (although some saints end up with attributes of local gods or get confused with mythological characters of the same name, like St. Hippolytus, who was a real person from Rome, but the tradition concerning his martyrdom comes from a mythological Hippoltyus of Athens).

In the Hellenistic world, there was a bit of a trend towards monotheism among Neoplatonists, proto-Gnostics and others, but they tended to find the idea of a personal God, who directly creates and participates in the world somewhat bumpkinish and backwards. Philo of Alexandria, a Hellenized Jew, worked extensively to marry Jewish theology with Greek philosophy.

Tacitus the Roman historian, on the other hand, thought that the Jews were peculiar for worshipping Jupiter alone, thought they came from the region of Mount Ida in Crete (because in Latin Judea was Iudea) and that Jerusalem (Hierosolyma) was named after King Solomon.

As a further note to /u/Gama_Rex and /u/Borkton it was normal for the Romans to insist that subject people make sacrifices to Roman gods for the health of the emperor or on moments of civic import. The Jews were exempt form this requirement because the Romans knew that there would be a huge fuss if they forced the Jews to commit idolatry by sacrificing to a Roman god. Personally, I think that they did not insist on it because they thought (probably correctly) that forcing the Jews to sacrifice would lead to a revolt.