d4 opening

What is the best defense to play against a d4 opening? I noticed that Bobby Fisher played the King's Indian Defense a great deal, but what is best for an average player like myself?

Dealing with absolutes like "best",  "worst" tends to simplify a  game that cannot be simplified, period.  There is no such thing as best and worst when it comes to a choice of opening. If there was,  nobody would play "second best" now, would they?

 Maybe a certain move in a certain line, but not when it comes to a choice of openings as broad as "vs 1.d4".   A lot of the different ideas transpose into one another anyhow.

You ought to find an opening system that works for you and helps you reach a playable middle game against the competition that you want to play against.  That's ALL an opening ever does ... it really doesn't win games for you at this level.

With that being said, the KID is a good system to learn no matter if you're playing with or against it.  

I'd say it's worth doing a tour of duty with a typical Queen's Gambit declined line given how much of a learning experience it can be in terms of playing against 1.d4.  I"d recommend the Lasker Defense ... nice + easy to understand ideas for Black.

The KID might requires a lot of theory and even if you learned a lot of variations in the king's indian, then the chances are that at your level your opponents will step outside the theory before move 4.