The blue string you are referring to is called a techeiles string, after Numbers 15:38 where it states:
ויאמר יי אל משה לאמר: דבר אל בני ישראל ואמרת אלהם ועשו להם ציצת על-כנפי בגדיהם לדרתם ונתנו על-ציצת הכנף פתיל תכלת:
My translation: And Hashem said to Moses, "Speak to the Children of Israel and instruct them to make for themselves tzitzis on the four corners of their garments throughout their generations and place upon each corner a cord of blue in each tzitzis."
The cord of blue is talked about in the Talmud (source?) in which it explains this blue cord's dye comes from a certain sea-creature from the Mediterranean Sea called the chilazon (חלזון) which surfaces once a lifetime (every seventy years in the language of the Talmud). By the time the Talmud was compiled, however, this mysterious recipe and sea-creature of the blue dye became lost, and tzitzis started being made with all-white strings. The blue dye is called "techeiles" in Hebrew. (תכלת)
For years this dye was lost, but in recent times, some authorities have claimed they have rediscovered the chilazon and have extracted blue dye from it. There are three opinions: the chilazon is lost, and it will not be rediscovered until Eliyahu Hanavi shows it to us in the Days of Moshiach, the cuttlefish is the chilazon, and the crustacean Murex Trunculus is the chilazon. Let's talk about these opinions.
The first opinion is that the chilazon and the recipe for techeiles is lost, and will not be rediscovered until the Days of Moshiach when Eliyahu Hanavi points it out to us (source?), which goes with the writings of the Holy Ari. This is the commonly accepted opinion, so most tzitzis you see people wearing are all-white.
The second opinion is that the cuttlefish is the long lost chilazon. The last rabbi of Radzyn, after studying much in the Talmud and visiting an aquarium in the late 1800s, formed the opinion that the cuttlefish was the chilazon, and successfully extracted blue dye from it. His chassidim began wearing it in their tzitzis, and most Breslover Chassidim also wear it in their tzitzis as well. This opinion, however, is not highly respected, and is generally turned down because of many reasons, including the discovery of an ancient piece of Judean cloth which contained a blue dye coming from a sea creature discovered in 1995 called the Murex Trunculus.
The Murex Trunculus is a type of crustacean that lives in the Mediterranean and has been widely accepted by many as the true identity of the chilazon, after much research and debate by the Ptil Tekhelet Organization, discoverer of the dye recipe. Because the dye on the ancient cloth matches the techeiles produced by Ptil Tekhelet, and other archaeological finds match as well, this dye holds most ground for being the real techeiles, which is why people don't really believe the Radzyn techeiles is the real deal.
Others say that the Murex Trunculus could be the real chilazon, and the dye produced by Ptil Tekhelet is real, but choose to continue the tradition of wearing all-white tzitzis on the basis that if a tradition is discontinued for 100 years, it becomes discontinued forever. But in short almost no one believes that Radzyner techeiles is real techeiles anymore. Even some Radzyners and Breslovers are now turning to Ptil Tekhelet.
The argument for the wearing of techeiles is a large topic, and cannot be discussed in great detail on Mi Yodeya. I will provide some further reading so you can learn some about it on your own. Forgive me also for my lack of sources.
Ptil Tekhelet Organization Website
Ben's Tallit Shop: Ptil Tekhelet and Radzin Techelet
Ptil Tekhelet: Introduction to Tekhelet
Also, slightly off-topic: I know for fact that Rabbi Chaim Richman of the Temple Institute wears Ptil Tekhelet, because of the last time I saw him. Some people believe that Moshiach is working around us now, and that the discovery of the chilazon ushers the Days of Moshiach. But it's your choice and most of the times (I say most of the times) people do not care either way, unless you live in Satmar maybe...