Within Europe, the standouts are Switzerland, Sweden and Spain. Ireland and Portugal also remained neutral. Other countries - Belgium, Holland, Denmark, Norway, would have liked to remain neutral but weren't given much choice in the matter.
Switzerland and Sweden were the subject of a thread a while back: http://www.reddit.com/r/AskHistorians/comments/1dex7j/how_did_sweden_and_switzerland_avoid_occupation/ - the basic answer is that both would have been difficult conquests for no real gain and particularly in the case of Switzerland they were probably more use neutral for their banking system.
Spain was generally sympathetic to the Germans, but having just come out of a draining Civil War were in no shape to officially join the Axis. However, Spanish port facilities were unofficially made available to U-boats and a volunteer Spanish division - the Division Azul - was formed and sent to help fight Communism on the Eastern Front.
Ireland had nothing really to gain from being involved, and Pro-British sentiment was not exactly high in most of the country. There were some extremists who believed that a German victory would mean the British being forced to give up Northern Ireland as well, and the Germans seem to have armed and encouraged the IRA as a way of tying down British troops. Again, a controversial topic and one where you'll find references and arguments either way.
Portugal I know less about - they were part of the Allied camp in WWI and Portuguese troops fought on several fronts including in France. Portugal is Britain's oldest ally (since the 14th century) and I believe the alliance was invoked to give the Allies access to bases in the Azores, but that was as far as they were going to go. I'd be interested to know more about this as my Portuguese history isn't great!
Vatican City--created in 1929 by the Lateran Accords--remained neutral during WWII. The reason why they remained neutral is Article 24 of that same treaty:
In regard to the sovereignty appertaining to it also in international matters, the Holy See declares that it desires to take, and shall take, no part in any temporal rivalries between other States, nor in any international congresses called to settle such matters, save and except in the event of such parties making a mutual appeal to the pacific mission of the Holy See, the latter reserving in any event the right of exercising its moral and spiritual power.
The Vatican City shall, therefore, be invariably and in every event considered as neutral and inviolable territory.
For more on the Vatican during WWII, an earlier post of mine may be interesting for you.