Q:

PNP/NPN - Why do we not just use one of the two types?

Using PNP instead of NPN merely requires one to make circuit upside down and reverse current direction, if this is true then why don't we just do everything with NPN transistors? Are there any applications where we can only use one of the two types? Why?

A:

There have been times when transistors with specific characteristics were feasible in NPN, or PNP, but not vice-versa. 
Two examples : 
The 2N3055 power transistor was NPN, at a time when PNP transistors of the same power were simply not available (115W I think, later pushed to 150W). 
That led to the "output triple" - a combination of 3 transistors that could replace either NPN or PNP output transistors, with the power handled in either case by the 2N3055. This was used in the famous Quad 303 amplifier of the early 1970s.
Another example : the BC214 PNP transistor could achieve lower noise in a microphone amplifier or audio input stage than contemporary NPN transistors.
With further developments in process technology, these considerations are less important than they were.

A:

For many years, we did use only NPN transistors.  Well actually, they weren't called transistors but "vacuum tubes".  It would have often been convenient to have the complement of NPN or N-channel vacuum tubes, but those didn't exist.  So yes, it's possible.  You could do similar things today by designing a audio power amplifier, for example, using only NPN or only PNP transistors.
However, having complementary parts can be very useful and allows for circuit topologies not possible using just one polarity.  Think of something like a audio power amplifier that needs to drive its output equally above and below ground.  Individual transistors pull in one direction, and the rest of the circuit has to pull in the other direction if you want the signal to go that way.  It shouldn't be hard to imagine that you can use one polarity transistor to pull high and the opposite polarity to pull low.  The tops and bottom halves of such circuits are then mirror images of each other about ground, which requires flipping the polarity of the transistors.  The output stages of most transistor audio power amplifiers exploit this NPN/PNP symmetry and really do work this way.

