What are the advantages and disadvantages of a monoplane, biplane and triplane?
One way to look at the differences between them is by looking at the effects of having multiple wings.


Having multiple wings means that the same aircraft can produce more lift for the same wingspan, as they have a greater wing area. 

As a result, the biplanes and triplanes have lesser wing loading, which results in shoter to/landing and lower stall speeds.

The shorter wingspan results in reduced roll inertia and roll damping, which results in better roll rates in the bi- and tri- planes. This improved maneuverability was the reason for the surge in triplane designes during WWI.
Due to interference effects, the bi- and tri- planes actually require higher wing area to produce the required lift. Also, the drag is also higher than the monoplane, which gets more pronounced as the speed increases.
The main reason for the use of bi- and tri- planes was that they provided the required strength for the type of airfoils used. As high strength materials became available, this mattered less and the aerodynamic disadvantage of the biplane (and triplane) designs came into fore. 


The aerodynamic advantage of the monoplane along with improvements in materials used has made the monoplane the de-facto standard except for certain specialised applications (like low speed aerobatics)