How Does the Church Recognize Saints?

Canonization, the process the Church uses to recognize a saint, has only been used since the tenth century. It is important to note that canonization does not "make" a person a saint; it only recognizes what God has already done. 

Honoring the saints was part of Christianity from the very beginning. This practice came from a long-standing tradition in the Jewish faith of honoring prophets and holy people with shrines. The first saints were martyrs, people who had given up their lives for the Faith in the persecution of Christians. By the year 100 A.D., Christians were honoring other Christians who had died, and asking for their intercession. For hundreds of years, starting with the first martyrs of the early Church, saints were named by public acclaim. Though this was a more democratic way to recognize saints, some saints' stories were distorted by legend and some never existed. Gradually, the bishops and finally the Vatican took over authority for approving saints. 

The process begins after the death of a Catholic whom people regard as holy. Often, the process starts many years after death in order to give perspective on the candidate. The local bishop investigates the candidate's life and writings for heroic virtue (or martyrdom) and orthodoxy of doctrine. Then a panel of theologians at the Vatican evaluates the candidate. After approval by the panel and cardinals of the Congregation for the Causes of Saints, the pope proclaims the candidate "venerable." 
The next step, beatification, requires evidence of one miracle (except in the case of martyrs). Since miracles are considered proof that the person is in heaven and can intercede for us, the miracle must take place after the candidate's death and be the result of a specific petition to the candidate. When the pope proclaims the candidate beatified or "blessed," the person can be venerated by a particular region or group of people with whom the person holds special importance. 
Only after one more miracle will the pope canonize the saint (this includes martyrs as well). The title of saint tells us that the person lived a holy life, is in heaven, and is to be honored by the universal Church. 
Though canonization is infallible and irrevocable, it takes a long time and a lot of effort. So while every person who is canonized is a saint, not every holy person has been canonized. You have probably known many "saints" in your life, and you are called by God to be one yourself.