According to Catholic Online it is the local bishop who is first approached. The person to be considered must have been virtuous, except for a martyr, and follow the orthodoxy of doctrine. The bishop may then send the cause on to a panel of theologians at the Vatican, if the bishop is inclined to do so. The person is first considered venerable. This normally has a waiting period of at least five years after the person being considered has died. This removed much f the emotional appeal and makes the decisions more objective.
The next step is proof of a miracle that occurred as the result of intersection by the candidate. To clarify things here, people pray in adoration only to God, but may offer prayers of petition through saints and those who are at least venerable in the hope that the petitioned intercedes with God on their behalf. Care must be taken here, for if a new medical technique is being tried, or if the petitioner also prays to God Himself or a canonized saint the miracle would not be ascribed to the candidate for sainthood. There must be no other possible explanation of the miracle, and no other petitioned for the miracle.
If the person who has been declared venerable can be shown responsible for successfully interceding with God for a miracle, except for martyrs, the person is considered to be beatified. The process continues requiring a second miracle, then the candidate is presented to the Pope, and only the Pope can canonize. Because the process carries infallibility the Pope can either accept or deny the candidate. The infallibility of the Pope only applies to when he speaks officially on matters of faith or morals, but canonization is considered one of those times.
Actually, only God can make a saint, the above process merely recognizes what God has done.