If an egg or hatchling has passed the "magic point" at which it can grow up but does not have enough clicks/unique views/views to do so, it is called an emergency egg/hatchling. The "magic point" is currently 3 days 23 hours left, so any egg or hatchling which has this much time left or less can be considered an emergency and added to Emergency Rooms (ERs).
Varying Definitions and History
The "magic point" has changed several times in the past. This is why some people consider different time markers as ER.
Originally, the "magic point" was 3.5 days left, so anything below 3.5 days would be an ER egg/hatchling.
Then the "magic point" was changed to be a random time between 3 and 4 days left, individual to each dragon. Since dragons with more than 3 days left could still be lacking time, not views, only eggs and hatchlings below 3 days were considered ER.
With the most recent change of the "magic point", some people now consider anything below 4 days left as an Emergency, others still keep to the 3 days left, which is why ER-sections in fansites have differing limits. (ER-sections usually generate a lot of views very fast, so any egg/hatchling that isn't ER would probably get sick. Therefore they are usually programmed not to accept anything that has more than a set time left.)
In trades, ER eggs and hatchlings are considered more valuable than non-ERs, because the new owner doesn't have to wait before they grow. They can simply add their new egg/hatchling to a few ER-sections and it will hatch/grow up in very short time (as soon as it has enough clicks/unique views/views). Since time and scroll-space is a limited resource in DC, ER commons are often traded for freshly bred rares.
(For the egg/hatchling to still be an ER when the new owner catches it, it needs to have been an ER for at least 24 hours if it has not yet gained 24 hours on the AP.)