Tu B'shevat literally means the "15th of Shevat" the Hebrew month which falls in the common calendar's January to early February. But Tu B'shevat is considered the Jewish New Year of Trees. Trees are extremely important in Judaism: trees are used metaphorically to consider G-d and life in Judaism and the planting of trees is holy. Combine that with the very modern importance of reclaiming the baron desert of Israel through planting trees, and it's easy to see why a celebration of trees is part of our tradition.
(There are actually four "new year" celebrations in the Jewish calendar: Rosh Hashanah being the one we all know and love for the apples dipped in honey. There is also 1st day of the month in which Passover falls which is the New Year of Festivals, and then the first of Elul, falling late August, is the New Year of Cows. Really. )
The origin of Tu B'shevat
The Jewish New Year of Trees was originally the date Jews used to calculate the age of trees. A tree is as old as how many Tu B'shevats it has been on Earth. Why is this important -- to calculate the age of trees? Well, there is a Commandment in the Torah that people shouldn't eat the fruit from trees for the first four years: no one eats for the first three and then during the fourth year the fruit is picked but not eaten (instead is donated to "G-d"). Then after the tree reaches four years of age, people can eat the fruit. Go ahead and ask "why four years?" but there is no good answer except "it is Commanded."
The Tu B'Shevat seder began as a Kabbalistic Jewish mystical practice in the 17th century. Ecologically minded Jews have adopted Tu B'shevat as a time to honor conservation and sustainable agriculture. (I bet Madonna celebrates.) The seder pretty much consists of drinking wine, eating fruit, praising G-d for creating these things, and reminded ourselves that we are stewards of the Earth. One way we drive home the point is by eating carob. Carob is indigenous of the land of Israel and it takes a special meaning because of how long it takes to bear fruit: up to 70 years. "Why plant a carob tree if you won't live to eat its fruit?," an old man is asked in a Talmudic tale. The old man replies that it is for his children and grandchildren. We must take care of this world for the next generation.
Along with carob, it is traditional to eat the "seven species", which are fruits of the earth mentioned in the bible and native to the land of Israel: pomegranates, dates, barley, wheat, figs, olives, and grapes. Also, participants get to drink four glasses of wine ranging from white to red. So, for example, you'd start with a Chardonnay, moving to a White Zinfandel (those morally opposed to white zin can obstain), continue with a Pinot, and end with a Cabernet.