Well, there's a difference between praying and worshiping. Here's a brief explanation.
It's a similar sort of thing to what happens on Earth. "I've got cancer, can you pray for me?" However, we believe that, because the Saints are in heaven, with God, if we ask them to pray for us, it'll be better! Mary, as the Mother of God and without sin, is even better.
We don't pray to the Saints or Mary as a divinity. That would be pretty bizarre and heretical. However, we ask the saints and Mary to pray for us.
Okay, that makes enough sense in context I suppose and amounts to basically what I expected. Now: do Catholics think it actually makes a difference to ask saints to pray for them? Why? Shouldn't the results of prayer be dependent on what they're asking for and who they are? Seems a bit peculiar that God might change His mind just because a saint thinks it's a good idea too.
I guess that kind of gets into the question of why prayer matters at all of course, since it probably shouldn't be possible to change God's mind about anything anyway, but that's a deeper theological thing that was already mentioned in the other Christian thread I think. Don't recall if it ever got a response though.
The real question here, as I see it, is about prayer. That's really covered in the Bible. The Parable of the Unjust Judge and the one about the man whose friend was visiting.
The parable of the Unjust Judge says 'If you could get an unjust judge to give you good things by pestering, how much more your loving Heavenly Father?'
The other one.
"He said to them, "Which of you, if you go to a friend at midnight, and tell him, 'Friend, lend me three loaves of bread, for a friend of mine has come to me from a journey, and I have nothing to set before him,' and he from within will answer and say, 'Don't bother me. The door is now shut, and my children are with me in bed. I can't get up and give it to you'? I tell you, although he will not rise and give it to him because he is his friend, yet because of his persistence, he will get up and give him as many as he needs."
That one is more about the saints. It refers to the relationship between God and the Saints and You. Take an earthly example. If you want the President to do something for you, do you ask him straight, or do you ask his friends (who happen to be your friends) to ask him?