Is constant air pressure maintained inside the fuselage?
For the purposes of air pressure inside aircraft, they are broken down into two broad categories:


Unpressurized - These aircraft are typically piston powered and can't go very high, so do not even attempt to regulate the air pressure inside the fuselage.
Pressurized - These aircraft do maintain pressurization in the fuselage, but it isn't constant beyond a certain point.


The category determines the answer to your question:

Unpressurized Aircraft

Unpressurized aircraft are not built air-tight for a couple of reasons:


It is very hard to make something as large as an airplane truly air-tight, and keep it that way.  Over time, seals will start to leak, fittings will loosen, etc.
The fuselage would have to be much stronger to withstand the pressure difference between the inside and the outside.  This would make them heavier and more expensive.
There isn't much need.  These airplanes don't fly high enough to really require a pressurized fuselage and a person can breathe more or less normally even though you aren't at sea level.  (Think of people that live on mountains.)


Pressurized Aircraft

Without going into the details of how an aircraft is pressurized (which isn't really relevant to your question), the air pressure inside the fuselage of a pressurized aircraft is usually maintained at sea level pressure up to a certain point.  However, the fuselage is designed to withstand a particular maximum differential air pressure, so in order to protect it (basically keep it from "popping" when the air pressure outside the airplane drops too low), the pressure is allowed to decrease when needed. 

In typical jet airplanes, the pressure inside portions of the airplane (typically the passenger compartment, some of the cargo holds, and maybe an avionics bay) at cruising altitude is maintained at a pressure which corresponds to an outside air pressure at an altitude of about 5,000 - 8,000 feet (referred to as "cabin altitude"), depending on the cruising altitude and the design of the airplane.