I believe there are saints in other sects of Christianity and even in non-Christian religions. However the Catholic Church is probably the most well known for its saints. So that's what I'm going to go off of here. There's "saint" and then there is "Saint". To be a saint with a little s, congratulations! Everyone in heaven is a saint. (The case is also occasionally made for little s saints who are still alive, but that's a tad bit more complicated)

In order to be a Saint, the process is a fair bit more complicated. You can't nominate yourself to be a Saint, actually for that matter, you can't be alive and be a Saint. There's a lengthy process that begins after your death.

First there's a waiting period, it's supposed to be 5 years so that people can objectively evaluate you. The people who knew you will be playing an important part later in the process, so it's important that they can look at your life objectively. However the Pope can waive this period if he so desires, Mother Theresa is the example that springs to mind regarding that.

Second, once your five years are over, or the Pope decided that the five years weren't necessary in your case, the bishop of whichever diocese where you died can begin an investigation regarding your potential Sainthood. The basic things that are looked for in this investigation are holiness and virtue. This is where the people who knew you come in as witness testimonies, as well as other evidence about your life and what you did in it, are gathered. If the bishop decides that there is enough evidence, he contacts the Congregation For The Causes Of Saints. This is the group that basically recommends to the Pope various people for Sainthood. The bishop then asks the Congregation to open the case for your Sainthood, if it's accepted by them for consideration, congratulations, you are now a "servant of God".

The next step is to have the Congregation take a careful and deep look into your case. They look for a number of things, including how holy you were, what work you did in your life, and most importantly, if people were drawn to pray based upon your example. If the Congregation approves the case, they pass it along to the Pope.

The Pope then takes a look at your case and decides if you did or didn't live a life of "heroic virtue". If the Pope does in fact decide you lived a life of heroic virtue, congratulations again! Because now you are given the title of "venerable".

The next part of this process is known as beatification. In order to ascend to this level, prayers made to you need to result in a miracle. If the prayers are granted, then it's proof that you're already in heaven because you and God are now working together on other's behalf. These miracles need to be verified and until they are, they are simply known as incidents. After the miracle is verified you are bestowed with the title "Blessed". The only exception for this is if you died for your faith, then you can be beatified without a verified miracle.

The last step in the process is called canonization. Canonization occurs when another miracle is attributed to someone who has already been beatified. The ceremony for canonization is actually fairly simple. The Pope performs a special mass, after which the person is considered canonized. It's important to note that canonization doesn't actually make a person a Saint, but it's more of an official recognition, that is infallible and irrevocable, that person is in heaven, lived a holy life, and will be honored by the Catholic Church.

Edit: That's the modern way to become a saint, Saints were originally just democratically proclaimed to be Saints, by the 900s, the Vatican had taken control of the process to approve Saints. Then in '83 John Paul II reformed it to the modern way.