Onsen Egg - or more commonly known as Onsen Tomago - is a way of slowly cooking your eggs in warm hot springs to get silky smooth whites and a beautifully runny yolk.
In Japanese, Onsen means 'hot springs', and tamago means 'egg', so it directly refers to the way the egg was cooked. The eggs are poached inside their shells, so while they are almost the same as soft boiled eggs, the whites are actually soft and silky, as opposed to firm whites, and the yolk comes our firm but creamy, as opposed to soft and runny.