Although commonly referred to as text or "ASCII"-graphics, DF uses a bitmap tileset with charactes from CP437, displayed in 16 colors. Both, colors and tileset can be changed (see below). In addition, interface text can be displayed with a TrueType font and creatures (which normally are displayed as letters) can be assigned to separate tilesets called graphic sets. The main tileset is sometimes called "character tileset", while the graphic sets are called "object tilesets".
Often, tileset creators offer preinstalled downloads or folders you just have to drop into your DF folder and overwrite files when prompted. They usually come with installation instructions either in a readme file or in their respective forum thread. In addition, there are various launcher applications that let you install and change graphics automatically. For manual installation of the various components, see here:
Put the tileset you want to use into the data/art/ folder. Open up init.txt (in data/init/) with a text editor and change the entries FONT, FULLFONT, GRAPHICS_FONT, and GRAPHICS_FULLFONT to the filename of your new tileset.
Put the graphic set into a subfolder in raw/graphics and the corresponding text file directly in raw/graphics. If you have an active save you will have to put them into the raw folder of your save, as well (data/save/<your region>/raw/graphics). Finally, set GRAPHICS to YES in data/init/init.txt
Replace colors.txt in data/init with your new colors.
True Type Font
Replace font.ttf in data/art with your new font.