If you are collecting them for eating, you can put them in the frig immediately. Some people feel that eggs can sit out without refriging them for a week, but I go the safe route with eggs and refrig them right away. If you are collecting them for incubation, you do not put them in the frig. They are set on the counter at temps around 50 to 60, give or take degrees, for a few days, turned occasionally, and then set in a prewarmed incubator.
An egg is laid with a protective coating, known as a "bloom", which allows air in but blocks bacteria from getting into the egg. As long as you don't wash the egg, it should be fine on the counter for a couple weeks. If you wash it, the bloom is washed off and it needs to be refrigerated. The egg can last for up to a month in the fridge. We always float the eggs in a glass of water before we eat them to make sure they are still good. If the egg sinks or stands up at the bottom, it's ok. If the egg floats (even the slightest bit) it's spoiled.