Prestige allows you to gain noble ranks which allow you to get better heroes, more heroes to attack with and new abilities. There are several ways to gain prestige.
Attacking members of other familiesEdit
This is probably the most common method of gaining prestige. Open the world page and select an area that is a different color from your family. Select view. You will notice that the names have different colors. These represent how much a player has been attacking an opposing family. The colors follow the standard color scheme. Attacking a player with a higher ranked color gives more prestige. However generally the people who can carry out many successful attacks are not easier to beat. To start attack some lower ranked people and work your way up. Noble rank and level are also fairly accurate ways to detect if a player is easy to beat or not.
This can get quite expensive as it involves spending large amounts of silver. Open the world page and select the area that you are in. Click on the investment tab. Select the amount to invest. Although the amount of prestige given is based on your level and the level of the world you are in the amounts do not curve. Investing in the smallest amount ten times is the same as investing in the largest amount which costs ten times more. However the is a limit to the number of times you can invest.
At the Noble Rank tab you can invest exploits to get a minor amount of prestige through that.
Certain levy events can cause you to gain more prestige. Check this google document for specific information about which event does what.
The league tech of Rumor can increase the growth of your prestige, meaning that you get rewarded more prestige from attacking enemy league members.
Competing in championships battles can also earn you pristige. And, if you get a large enough kill streak each battle can start awarding much more prestige (at 12 i was getting 450-500 each battle).