Dragon growth is based on several different factors: Views, Unique Views, Clicks and Time.
Views denote the overall number of views your egg or dragon has received. This can be misleading as the amount of views you can receive is restricted by the amount of unique views you have. For every unique view your dragon accumulates, it can receive a maximum 15 views. So, if an egg has 10 unique views it can't have more than 150 views. Views, some may argue, are a key decider on when a dragon will hatch or grow up, although no formula has ever been released by Dragon Cave administration.
Viewing your own scroll won't increase your egg's view count, but viewing the scrolls of others will increase their eggs' and dragons' view count. Going onto a dragon's view page will give one view, but refreshing the page won't give more views. Refreshing the scroll page however, will keep increasing the view count until it reaches its ratio limit. If the ratio of view:unique view approaches or hits the 15:1 limit, then the egg can undergo Dragon Sickness.
Tinsel Dragons, Shimmer-scale Dragons and Avatars generally take longer to hatch, and get sick more easily. Prize dragons generally take about 5000-12000 views to hatch and grow up. An egg will get sick if it gets too many views in a short space of time - try to keep it a moderate amount.
Unique views are accumulated every time an individual views your eggs. However, unlike views, you can only gain 1 unique view per IP Address. Unique views are important since they decide how high your views count can go. As previous mentioned, normal views can be no more than 15 times the number of unique views. One unique view can be gained for each IP address viewing the dragon. Unless you have an IP address that changes or you move about and change your IP address, you can only give one unique view to every growing dragon you view. It is understood that the more unique views a dragon has, the less views it needs to mature.
Around roughly 400 unique views, a dragon will hatch, although for the rarer dragons, the amount needed may be drastically larger. Roughly 1,000 unique views are needed for maturing. It is possible for a dragon to hatch and grow with a lot less then this (for example an adult dragon only having around 600-700 unique views). However it isn't recommended as they have a higher chance of dying.
Usually, your egg starts cracking at around 130-400 unique views, and continues to crack with more views.
Clicks, like unique views, can only be given out once per person. A click happens when someone accesses a dragon's statistics or "View" page. Clicks are not essential to a dragon's growth, as a dragon can grow without any clicks at all, but the presence of numerous clicks speeds up the growth process. Like unique views, only one click can be given to a growing dragon per IP address. The more clicks a dragon has, the less views and unique views it needs to hatch or mature.
Gaining views/unique views/clicksEdit
There are many ways to get views, unique views and clicks for your eggs and hatchlings. Many people use a "Click Site" to get the necessary views. Another way is to put them in a forum signature using the BBCodes shown in the "Show HTML/BBCode Links" link on the egg or hatchling's view page (since August 17th, 2015, the "Direct Link" option has been removed from this page). Both ways are equally fine, though using click sites has become the more popular way to go.
Even though your egg may have plenty of views and has a big hole in it, or your hatchling has plenty of views and has matured, hatching and maturing are still dependent on time. Until your dragon reaches its "magic time", nowadays known as "ER or Emergency Room Time", it cannot move on to the next stage. This magic time has shifted in the past, originally being 3.5 days, then shifting between 3 and 4 days, and becoming individual for every dragon. Currently, the magic time for all eggs and hatchlings is just under 4 days, and is now known as "ER" or "Emergency Room Time".
If your dragon has reached its "time" but has not yet reached a high enough views/unique views/clicks count, it will not hatch or mature until it reaches this count. If the dragon does not meet the required number of views by the time it reaches 0 days, it will die. This is why eggs and hatchlings with fewer than 3 or 4 days (depending on personal opinion) remaining are considered emergency dragons, with click sites setting up "Emergency Room" sections where those ready to hatch can receive the maximum amount of views and clicks in the shortest amount of time as they are ready to grow.
There are dragons whose hatching time is linked to a specific bracket of Cave Time, and these dragons will not hatch until this bracket of time has been reached. Check your dragon breed's page for information about this.