Dragon growth is based on several different factors: Views, Unique Views, Clicks and Time.
Views
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 Dragcave administration.
Viewing your own scroll won't increase your egg's view count, but viewing the scrolls of others will increase the 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 will undergo Dragon Sickness.
Unique Views
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 was stated in the previous section, 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. 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
Clicks, like unique views, can only be given out once per person. A click happens when someone accesses a dragon's statistics 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 mature.
Gaining views/unique views/clicks
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 views necessary. 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.
Time
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 grown wings and gendered, hatching and maturing times are randomized. Until your dragon reaches its "magic time", it cannot move on to the next stage. This magic time has shifted in the past. It used to be 3.5 days, then it was a random point between 3 and 4 days, individual for every dragon. Currently the magic time for all eggs and hatchlings is about 4 days.
A dragon that has enough views to do so will generally hatch/grow up when its counter reads "4 days.". 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.
However, 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 does reach said 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.
When submitting to The Allure Of Neglected Dragons, Valley Sherwood or Dragon Dump, there is a counter below your dragons telling when the next egg or hatchling can hatch/grow up.
Notes
Tinsels, Shimmers, and Avatars may take longer to hatch, and get sick more easily. Tinsels 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.