Tu B’Shevat, the 15th day of the Jewish month of Shevat, is a holiday also known as the New Year for Trees. The word "Tu" is not really a word; it is the number 15 in Hebrew, as if you were to call the Fourth of July "Iv July" (IV being 4 in Roman numerals). See Hebrew Alphabet for more information about using letters as numbers and why the number 15 is written this way.
As I mentioned in Rosh Hashanah, Judaism has several different "new years." This is not as strange a concept as it sounds at first blush; in America, we have the calendar year (January-December), the school year (September-June), and many businesses have fiscal years. It's basically the same idea with the various Jewish new years.
Tu B’Shevat is the new year for the purpose of calculating the age of trees for tithing. See Lev. 19:23-25, which states that fruit from trees may not be eaten during the first three years; the fourth year's fruit is for G-d, and after that, you can eat the fruit. Each tree is considered to have aged one year as of Tu B’'Shevat, so if you planted a tree on Shevat 14, it begins it second year the next day, but if you plant a tree two days later, on Shevat 16, it does not reach its second year until the next Tu B’Shevat.
Tu B’Shevat is not mentioned in the Bible. I have found only one reference to it in the Mishnah, and the only thing said there is that it is the new year for trees, and there is a dispute as to the proper date for the holiday (Beit Shammai said the proper day was the first of Shevat; Beit Hillel said the proper day was the 15th of Shevat. As usual, we follow Beit Hillel.
There are few customs or observances related to this holiday. One custom is to eat a new fruit on this day. Some people plant trees on this day. A lot of Jewish children go around collecting money for trees for Israel at this time of year. That's about all there is to it.
Sources: Judaism 101