Q:

What is the relationship between Fillory and Narnia? 

I've recently started reading the book The Magicians by Lev Grossman.  There is considerable mention of a land known as Fillory, which seems to be related to Narnia, from C.S. Lewis's series.. Is there an intentional relationship, and if so, what is it?  

A:

Yes.  From an interview with Grossman:

The funny thing is, Lewis was notoriously sloppy as a world-builder.  He liked to drag in whatever was handy — nymphs, fauns, wizards, Father Christmas, whatever — without much regard to internal consistency.  For Christ’s sake, Mrs. Beaver has a sewing machine!  It drove Tolkien crazy.  Part of the joke of The Magicians is that I’m taking a Narnia-style fantasy world and forcing it to behave consistently.  It turns out that you have to bend it and distort it and break it to make it fit. 

A:

The Magicians essentially takes a re-look at two of fantasy's most popular tropes - the magic boarding school and the children off to fantasy land.  The latter is exemplified by Narnia, and Grossman seems to have based Fillory off Narnia.  So you have a bunch of children who go to Fillory, and fulfill the quests provided by the god-sheep. 
However, as you read further you will find that though Fillory does resemble Narnia on the surface, it is given a much more adult treatment.  People still die, and they seem to have more realistic personalities etc.  The biggest difference is that unlike the books that follow the two tropes in a traditional manner, happiness is not guaranteed, and magic does not automatically make people happier or better. 

A:

Fillory plays the same role in the characters' childhoods and imaginations as Narnia does in our world, but it doesn't have a bunch of copyright lawyers potentially issuing cease-and-desist notices to authors who write about it.