Praying to a saint is asking someone to pray for you. That is the whole of it. You would ask a friend to pray for you before surgery, and you would not think you were worshiping the friend. The saints are friends who have finished the road, and the asking works the same way.
It rests on one conviction: that the people who died in Christ are alive in him, and still care what happens to us. The Letter to the Hebrews, after a long roll of the faithful who came before, says this:
And therefore we also having so great a cloud of witnesses over our head, laying aside every weight and sin which surrounds us, let us run by patience to the fight proposed to us:
Hebrews 12:1, Douay-Rheims
And when John is shown heaven, the prayers of ordinary people are there, carried up:
And when he had opened the book, the four living creatures, and the four and twenty ancients fell down before the Lamb, having every one of them harps, and golden vials full of odours, which are the prayers of saints:
Apocalypse 5:8, Douay-Rheims
So the saint does not answer the prayer. God does. The saint asks with you, and, as James says of the just, that asking counts for something:
Confess therefore your sins one to another: and pray one for another, that you may be saved. For the continual prayer of a just man availeth much.
James 5:16, Douay-Rheims
You say the name, and you ask. That can be your own words, or a prayer that has been said to that saint for centuries, or a novena, which is the same asking kept up for nine days. Nothing about it replaces praying to God directly; it sits alongside, the way asking your family to pray sits alongside praying yourself.
The Church's own summary is short: those who are with God do not stop caring for those still on the way, and their prayer for us is the greatest thing they can do. It is in the Catechism, paragraphs 956 and 957.
Jude for a hopeless cause, Gerard for a mother expecting a child, Dymphna for a mind in trouble. A saint is turned to for what their own life touched. Gerard was a Redemptorist brother whose care for mothers and children was known while he was alive. Dymphna was a girl who fled a father who had lost his mind. The pairing is not assigned; it grew, because people found that it fit.
So the way in is to find one whose life touches yours. The saints on this site are chosen for what they brought, with what to pray with each. The novenas carry the ones most often turned to. And if none of that fits yet, the simplest form is still the oldest: Saint N., pray for us.