Dio is Ronnie James Dio, the replacement vocalist for Ozzy Osbourne in Black Sabbath. His solo career was far less successful than Ozzy's, and there is a mutual hatred between the two.
This song is about Satan, and his descent to Hell (his "dive"). The lyrics deal specifically with Revelations 12:9 where Satan was cast to the earth. The tiger is symbolic of the wild beast that comes out of the sea, as mentioned in Revelations 13:1.
There are two characters in this song: the tiger and the holy diver. The tiger is the Sun God ("the cat in the blue is coming after you, holy diver") and the holy diver is an ordinary person who sees the hypocrisy of religion ("the velvet lies") and finds himself lost ("in the midnight sea" or darkness). There are "shiny diamonds" or stars as if "the eyes of the cat" were in both "the black and blue" or night sky as well as day.
The impulse here is to get away from religion ("something is coming for you") but still seek that guiding light ("race for the morning"). The paradox is quickly realized ("you can hide in the Sun until you see the light"): that which he is running from he is also searching for. So, the conclusion is to "ride," "jump on," mount the tiger, taking personal control of his beliefs and rejecting organized religion.
Generally speaking, things are not always what they seem. (Is the demon on the cover drowning the priest, or the priest drowning the demon?) The tiger looks dirty but there is nothing wrong with him. You can feel his love, but don't let your guard down. Finding balance is a recurring theme in Dio's works.