import instructions for cats and dogs are available here:
vaccines and paperwork needed are listed there
quarantine is required, at one of two locations:
 (south Iceland)
 (north Iceland)
it's not simple to move here with a pet- lots of paperwork, checks, and quarantine. I don't personally know anyone who's done it.
for small companion animals other than cats or dogs, some applications here:
anyone wanting to bring a livestock companion animal (horse, pig, goat) is probably out of luck. Iceland is very restrictive for such animals and even Icelandic horses who leave the country aren't allowed back in for fear of disease. Even riding and fishing equipment has to be dealt with specially when being brought in.