Today, T"U BISHBAT, is the fifteenth day (in Hebrew the letter ט/T=9 and the letter ו/U=6. T+U=15 ) of the month of Shebat, which is the new year, Rosh haShana, of the trees. In the fifteenth of Shebat the trees are considered one year older. Halakhically speaking, the age of a tree does not increase on the date it was planted (the tree's "birthday"); all trees become one year older on the same day, T"U Bishbat. This in an important factor for many Mitsvot connected to agriculture, for example, 'orla: the prohibition to eat the fruits of a new tree during the first three years. For this and other Mitsvot one needs to know the age of the tree (see more details of the application of T"U Bishbat in the Mitsvot here). By the way, unlike most Mitsvot related to trees and plants, which pertain only to Israel, 'orla is a Mitsva that applies not only in Israel but outside of Israel as well.
How do we celebrate the 15th of Shebat today?
There are no Halakhic instructions for any celebration of T"U Bishbat. The custom in many communities is to eat at night or during the day all type of fruits, and especially the seven fruits or species by which the Land of Israel was praised in the Tora: "...a land of wheat and barley, grape and fig and pomegranate; a land of olives and honey (=dates)" (Deut. 8:8).
Certain families have the custom to organize a Seder of T"U Bishbat. Saying berakhot ha'ets for the fruits of the tree and the correspondent berakha for whatever else is eaten, like wheat, barley (normally mezonot), etc. Many consider a special zekhut to have for this Seder fruits from the Land of Israel and say berakha for them.
No special prayers are added to the regular services today. Tahanun, however, is not recited in T"U Bishbat.