How a Saint is Made
The current process for declaring someone a saint was established by Pope John Paul II in 1983 in the document Divinus Perfectionis Magister. This document continued a process of simplification started by Pope Paul VI during Vatican II.
So how do you get to be a saint?
First, you have to be dead at least for five years. Yes, I know that your mom is a saint, but unless she is dead the Church isn't going to take an interest in canonizing her. (Canonization is the Catholic term for declaring someone a saint.) The pope can waive the five year requirement but don't expect an exception on the death part.
Second, you have to either a) die for the Faith (a martyr) or b) demonstrate "heroic virtue." Dying for the Faith may seem like a pretty remote possibility these days, but some of the most deadly persecutions against Catholics have happened within the last 100 years in Spain and Mexico, and in many places in Africa today being a Christian greatly reduces your life expectancy.
So for those of us who don't have a martyrdom to look forward to, what's heroic virtue? Fortunately, like in most things, the Church has divided things into nice categories. There are the theological virtues (faith, hope and charity) and the moral or cardinal virtues (prudence, justice, temperance, and fortitude). If you had to memorize the Baltimore Catechism you already knew that.
Exhibiting heroic virtue doesn't mean that you are perfect or look like one of those doe-eyed pastel saints on holy cards (ick). It does mean that you live a life so that anyone who looks at the list of virtues can say that you excelled at them all.
Start a Cult
Next, a bunch of people have to actually know that you lived a life of heroic virtue and decide to start pestering the local bishop about it. These people have the task of making a case to the bishop that you deserve some sainthood scrutiny and also of raising money to pay for the process. Yes, canonization isn't free and the Church doesn't foot the bill.
Once the bishop is convinced that you might be worthy of putting on the track to sainthood, an official proceeding is opened where as much information that can be gathered about your life is collected and gone over with a fine tooth comb. Those embarrassing pictures from college? That article you wrote for the newspaper? All will be seen by your local bishop and the committee that will review your life for sainthood.
Once the local bishop has decided that you did lead a life of heroic virtue, your cause is forwarded to Rome so that the Sacred Congregation for the Causes of Saints can go over all the information again. If the Congregation decides that you did lead a life of heroic virtue, their opinion is sent to the pope who decides if you should be declared "Venerable." If you make it this far, you are in luck. Most don't get this far and many causes have been in process for hundreds of years.
It Would Take a Miracle
A declaration of Venerable means that the pope agrees that you lived a life of heroic virtue but doesn't give you a spot on the Church calendar yet. Prayer cards asking both for the furthering of your cause and also for your intercession can be printed. This is important because the next step, Beatification, will take a miracle of the Miracle Max variety.
Avoiding a beating from dad or graduating from college doesn't count. This has to be a real miracle that doctors and scientists examine and can't find an explanation for.
For example, the miracle that was used for St. Elizabeth Ann Seton's beatification was the miraculous cancer cure of a nun. The other thing that is required to prove a miracle is that there is documentation that there was a pre-existing condition that has been cured through intercession. You can't say that you were turned into a newt and that you were cured through John Paul II's intercession unless you have some medical proof of your former newtness.
Once the doctors have determined that none of them could have cured you and your life is gone over again, the pope can decide to declare you "Blessed." Being a Blessed means that your local diocese or religious order can officially put you on their calendar and say masses in your honor.
You've Been Canonized!
Once you are a Blessed it's time to start waiting for another miracle unless more than one have been under investigation already. Once a second miracle is confirmed and a final review of your life is complete, the pope can decide to officially canonize you. This doesn't mean that you now suddenly get a pass into Heaven. The pope (or the Church) can't declare that someone is a saint who really isn't. The canonization is just a formal recognition by the Church that you are in Heaven and lived a saintly life.
This is it - you are now a saint and you get added to the liturgical calendar for the entire Church. Hopefully the artwork that has been done of you will be realistic at least and flattering at best. You probably wouldn't want to look like you are wearing rouge and photographed through a soft lens on the millions of holy cards that get printed of you. Ack.
For more details on the whole process we recommend How Saints Are Made, by Bill Dodds