calibro saidyes, you can, but it's a rabbi's job to turn you away three times before he let's you convert. the idea is that you have to want to convert to religion; unlike christianity, judaism isn't looking to convert people. you typically have to prove your worth before the rabbi will allow you to convert (this is in accordance to orthodox law)
This is absolutely right. Though I would like to add to this.
What your cousin said is wrong. Yes you have to either be born of a Jewish MOTHER (Judaism is matrilineal- you are what your mother is), it's a race and a religion. You can convert, but it is a sin to convert for marriage. And any Rabbi who sees that you are converting for marriage will not allow you to convert.
In order to convert you would have to immerse yourself in hours and years of talmudic study. The rabbi and the community would turn you away and deny you CONSTANTLY (traditionally at least three times) following the story of Ruth who swore to her mother in law "Wherever you go, I will go. Wherever you lodge, I will lodge. Your people shall be my people, and your God my God. Where you die, I will die, and there I will be buried" (Ruth 1:16-17).
When the Rabbi feels that you may be ready for the actual conversion he will have you meet with a beit din (a tribunal of riteous men) who will question you on your reasons, your knowledge, your faith, and your convictions. They will then perform a ceremony 9called the Brit Olam) to turn the scar from your previously non religious circumcision into a religious one. This is done by extracting a drop of blood large enough to be seen on linnen from the scar. If you are not circumcised, they will perform a circumcision. Yes the entire tribunal will watch. After you have healed from this, a minimum of 7 days later, you will take a ritual bath called a Mikveh in front of the Beit Din. You are completely naked in a body of water that is clear ad transparent to the bottom, deep enough to fully submerse yourself in, and is of a natural sorce (ie rain water or a spring, tap water does not work). Afterwards you will recite for the first time the Haftorah.
Once the process has been completed, it is illegal for other members of the faith to ever mention your life before your conversion or to mention that you were ever not a Jew. It is believed that a convert was always a Jew, just born to the wrong parents.
Below are some links for more information. If you are still curious, I can put you in touch with my Orthodox Rabbi who would be happy to answer more questions. Though he does not perform conversions, he's not comfortable with it.http://www.convertingtojudaism.com/http://www.chabad.org/library/article_cdo/aid/3002/jewish/How-Does-One-Convert-to-Judaism.htm