In video games, do they use images or do they draw everything using vertices?
The answer really is: both.
3D video games use collections of verticies to describe various models that are used in the game (scenery, player characters, etc.). Textures (images) are then applied to the meshes to make the game look realistic. Without the images, everything would look like a wireframe or a single color.
Some games use a technique known as "billboarding" which means they take a 2D image and render it directly in the 3D scene. The image is always facing the camera so as not to betray its 2D nature. Some games use a modification of this technique to simplify complex scenery like trees (they are referred to as "impostors"). The tree is made up of 2 or more 2D images as different angles to one another.
2D video games can still use verticies and render in 3D (but they restrict themselves to a single plane so it looks 2D). However, it's more common to use images (sprites).
That depends on the game and whether they are using directdraw or direct3d. DirectDraw only uses 2d surfaces, thus everything is an image, Direct3D however is built around everything 3d, thus to even draw an image you would draw a shape using vertices then apply a texture to it(the texture would be the image). It should be noted that directdraw is old(having stopped in dx7), and microsoft recommends using Direct3D with textured quads to do 2d blitting