What does gases at constant pressure mean?
Without a clearer setup it's difficult to say whether someone misspoke, whether you misunderstood precisely what your teacher was saying, or whether your teacher was incorrect.  

If the setup is that there is a single piston that contains two gases, and a weight is placed upon it, then:


The volume of each of the gases in that piston is equal to the entire volume of the piston.  The volume of any gas is the entire volume of its container.  If the gases are ideal, then the volume of the piston will be the sum of the volumes of the piston if each of the gases were kept in it individually.
The pressures of the two gases sum to the pressure exerted on the piston by the weight (and the outside atmosphere.)  The pressures will be constant as long as the amount of gas present, the applied pressure, and the temperature are kept constant.


If the setup is that there are two pistons that each contain a gas, and an identical weight is placed upon each (or a single piston filled with two gases, one at a time) then:


The volume of the two pistons will be proportional to the number of moles of ideal gas that it contains, as long as both are also at the same temperature (we're already assuming the same pressure due to the identical weight.)
The pressures of the gases in the two pistons will be equal, because they are opposing the same weight + outside atmospheric pressure.


As to your other questions:
Gases at different pressures do indeed exert constant pressure, assuming things like constant temperature, constant volume, and constant number of moles of gas.  The pressure of a gas is the result of huge numbers of gas particles striking the walls of their container every second.  The rate at which those particles strike and rebound and the force with which they do so determines the pressure.  Because there are sooooo many gas particles in any reasonable quantity of gas, the fluctuation in the rate and intensity of those collisions is nearly constant (to many, many decimal places.)

Gases at constant pressure do not necessarily exert equal pressures.  If I pump up the front tire of a bicycle to twice the pressure as the back tire, each has a constant pressure, but they are different.  If I put two moles of gas A and one mole of gas B into a sealed container, the pressure due to A will be twice that of the pressure due to B, but both will be constant, as will their sum, the total pressure of the container.