Which country is 岛国?
岛国 is used for Japan in most cases (especially on net), usually as a satirical usage (especially on net again). e.g.


  我们叫日本岛国叫韩国棒子，那外国人叫我们什么？


Why? I believe it's because among many geographical island countries, Japan is the nearest and biggest one, and having the most close relationship with China. Whether the relationship is good or bad, and how it changed and changes, Japan is the one China and Chinease most care about.

Here is a Japanese's introduction about this alias:


  中国人は確かに日本のことを「島国」と言ったりします。そして日本を「島国」と呼ぶときは、多くの場合において侮蔑的な意味合いを持ちます


中国人确实有时候将日本称为"岛国"。而且大多数时候带有侮辱的含义。
(Chinese people do call Japan as岛国sometimes, and contains insult meaning in most cases.)

Here're some other (more formal) names for Japan: 倭国, 东洋, 东瀛, 扶桑.