Under which condition does a country with a small gdp have a large per capita income?

GDP, or gross domestic product, is  how many goods and services a country or realm produces over an interval, usually a year.

the per capita part, means, the average per person, so you grab the GDP, say 5 trillion, and divide it evenly among all inhabitants, so a per-capita figure is really just GDP/population.

now, you can have a population like Germany with 80 million with a GDP of say hmmm 8000 million for simplification, the per-capita is then 8000/80 or  $100 per person.

now, you could have another population like say Japan, and let's just say is less people... hmmm 20 million, with a GDP of say 4000 million.

then the per-capita in that example for Japan will be 4000/20, which is $200 per-capita, notice, the Japan's example has a higher per-capita, though the GDP is smaller to the example of Germany, that's because the Japanese population is also smaller.

sidenote:  per-capita figures are often very misleading, since economies do not grow that linearly.