The viewfinder is an essential part of most cameras for previewing what might be the image after exposure. Only a few camera types are regularly not equipped with viewfinder, for example repro cameras. In digital cameras the viewfinder might be omitted since the LCD display on such a camera's back might serve as provisional viewfinder. It's regularly omitted on big plate cameras which are only used with ground glass focusing/previewing.
Waist level finders
In antique cameras the reflecting type viewfinder is the most common means of image preview. It is not very reliable but easy to add to a folding camera's front standard or folding bed, or into the housing of a box camera. The brilliant finder is the most widespread of such finders, a combination of a lens, a mirror at a 45 degree angle behind that lens, and another lens at right angle position to the first one to view the mirrored image from the top. An older version was the small cubic Watson finder with lens and mirror as in the brilliant finder, but with a hooded matte screen for viewing the finder image. A special version was the Sellar finder which consisted just of a concave mirror with targeting aid. Old SLR cameras have a bright reflecting type finder with matte screen that uses the same lens as the camera uses for exposures. Before exposure the mirror is lifted so that the light coming from the image subject through the lens can pass towards the image plane where the focal plane shutter allows the exposure of the film for an instant. TLR cameras have a bright reflecting type finder with its own focusable lens, a "twin" of the camera lens, combined with mirror and matte screen. Thus a TLR finder is almost like a camera obscura. All these finders are to be viewed from above. Together they are the class of waist-level finders (or chest-level finders, such names resulting from the height in which a camera is held when the finder is used).
Eye level finders
Other optical viewfinders of old cameras are placed upon the camera top as small rectangular "telescope". Since the 1950s these viewfinders were more and more integrated into the camera bodies. In older cameras the finders might consist of only the front lens and the ocular lens. The viewfinders' "telescope" optics are often more like reversed telescopes, giving an image of reduced size. These finders belong to the "reverse Galilean" or "Newton" type.
Modern zoom finders might be more sophisticated. More sophisticated are also viewfinders with superimposed rangefinders, and those with parallax correction. Parallax error is due to the fact that an optical system positioned parallel to the camera optics never gets the same image as the camera lens for close subjects. This can be corrected by moving the ocular slightly, or by some other provision to help make the finder image and taking lens image similar.
Modern viewfinders show more than the expected image. In most of these cases a so-called "bright frame" or "bright line" in the viewfinder indicates the expected exposure frame. Since 1960 several kinds of indicators were developed, the first were red/green indicators for correct- or under-enlightenment in the early cameras with selenium-meter controlled exposure. Nowadays a green LED is standard that's on when the autofocus finished focusing. Other information might be mirrored into the viewfinder or shown in an LCD section in the frame around the viewfinder image. In the 1970s and 80s a row of LEDs beside the finder image was common as scale showing the (proposed) shutter speed or a match-needle metering instrument replacement.
Finders of modern SLR cameras view through the taking lens and are completed with a pentaprism plus ocular or a mirror system plus ocular. With these additions the reflex finders become eye-level finders and enable the photographer to see the image like through a telescopic optical viewfinder, including any effects introduced by the lens itself or added to the lens.
A common old finder type is the frame finder, consisting of two frames, or one frame with or without crosshairs plus targeting aid. A big rectangular frame made of massive wire combined with targeting aid is called sports finder - as following fast action is comparatively easy with this type of finder, where the image is seen at correct scale, correctly oriented, and there is usually some visibility around the edge of the frame. A modern plastic variant of the sports finder can be bought as accessory for underwater cameras. A mixture of optical finder and frame finder was called Newton finder, with one lens and targeting aid.
A new type of the viewfinder is a combination of a miniature color LCD screen with an ocular lens. It's often found in compact digital cameras with a very wide zoom range.
Of course the inventiveness of camera developers didn't stop with the question "How can we combine the advantages of waist-level and eye-level in one finder?". One answer was a collapsible hybrid viewfinder. It had a front lens with crosshairs as a targeting aid, what was called Newton finder 100 years ago. With this combination it was an eye level finder. But when its crosshaired mirror was put in at a 45 degree position behind the lens it became a smart waist level finder. Smart because the crosshairs of the lens and that on the mirror helped to find the perfect perpendicular viewing position of the eye over the finder.