What advantages and disadvantages do tri planes have?
The accepted answer is wrong.

Triplanes have all the disadvantages of biplanes, only more so. But they share their advantages as well, and the main reason to build them in WW I was roll maneuverability:


The reduced wingspan reduced roll inertia, which helps to accelerate quickly into a high roll rate. Note that roll authority changes linearly with span (if the ailerons are scaled with the span), but inertia will change with span squared.
The reduced wingspan reduced  roll damping, so the achievable maximum roll rate is higher. The maximum roll rate is reached when the aileron-commanded rolling moment equals roll damping, and roll damping is proportional to span squared.


When the Sopwith Triplane became operational early in 1917, German pilots felt at a disadvantage and Reinhold Platz at Fokker created the famous Dr.I. The next generation were biplanes again, however, and the latest Fokker creation of WW I was actually a monoplane. The high climb rate of the Triplane was caused by its low wing loading and high power-to-weight ratio and more due to the lightweight construction and the advances in engines than the triplane layout.


  What were the advantages to having more wings?


Adding a second wing creates a lightweight, stiff box girder. As for why people added even more wings, see this excellent answer.


  And, why don't countries use them today?


Because countries don't use airplanes, pilots do.