I recently got the book "Chess traps pitfalls & swindles how to set them and how to avoid them" from the library and I noticed the chess notation in this is different. For example, one of the games he shows that continues from a diagram continues:

1...  N-B5  
2.PxN   BxN
3. N-N3 QxP  
4. NxB  
and it stops there. Another one is:

























1...  QxP
2.QxPch NxQ
3.R-Q8ch N-B1
4.R-R8ch KxR
5.RxN mate

Does anyone understand this type of notation? 

It's simply another type of notation.  I don't personally like it, but I guess it works for some people!

Yeah, it's the old notation.  Everyone uses algebriac notation now.  They call the squares by the pieces that occupy the file.

For example

e4 would be KP4 = King Pawn 4.

h4 - RH4 = Rook to H4.

It gets confusing when you have to label the bishops or knights.. instead of labeling them by the file, they label them by queen on kingside i.e.

KB= King Bishop or QB = Queen Bishop.  And the square is called King Bishop 4 instead of f4!  lol

It looks like what I know as descriptive notation. A number of older books have been updated from this to algebraic notation.

I recommend learning the old notation.  There's a lot of great old books and magazines that haven't been translated to algebraic notation (or you'd have to pay a lot for the new version).  You can find these great books at used bookstores, yard sales and the like.

I'm an old timer, so I remember when, in the 70s, Chess Life and Review started having more and more articles in algebraic.  I believe Fischer always used English Descriptive Notation.  Here's an example from 1970.  Boy was Fischer's handwriting illegible or what?

Fischer was, of course, quite adept in algebraic notation, consuming Soviet and other chess literature voraciously.