By default subscribed accounts can create 8 characters, while silver accounts can create 4. There are two ways to increase your limit:

1) Daybreak Store
2) Loyalty Tokens

Each one of these can increase your maximum character slots by 3, for a total of 6 additional slots. This means subscribed accounts can reach a maximum of 14 character slots, and Silver accounts can reach a maximum of 10.

There are some legacy methods that allowed even more character slots (Station All Access and Legends server), but these are no longer offered.

Daybreak Store

Open the Daybreak Store and go to All Items -> Services -> Slots. Here you’ll find an option to buy either 1 character slot at a time, or buy all 3 at once. Buying all 3 at once is a better value overall.

Buying 3 = 2000 DBC (1800 with +10% Discount if subscribed)
Buying 1 = 800 DBC (720 with +10% Discount if subscribed)

These may go on sale occasionally.

 Loyalty Tokens

 In Plane of Knowledge there is a vendor named Alerynril the Loyal. This vendor only shows up if House of Thule is released on your server. If House of Thule is not released on your server, then you can log on a live vendor to interact with her.

She sells the item Doll of Character for 1,296 Loyalty Tokens. When clicked, each of these will add 1 additional character slot to your account. You can buy a total of 3 then it will no longer be offered.

Loyalty Tokens apparently generate at the rate of 120 per week on an account with an active subscription. I did not personally verify this, although I believe it to be correct. At this rate it will take many months to have enough points to get all 3x Doll of Character. You can have a maximum of  5760 Loyalty Tokens. They can be found in the “Alt. Currency” tab in your inventory window.