Finder is window we look through to compose scene we want to photograph. In correlation with construction and camera type there are different types of finders:
Optical finder on compact cameras- Simplest finder construction solved in such a manner that above objective itself exists small optical element that, rather (in)accurately, tries to display what objective sees. The biggest downside of this system is displaying error because of the parallax and inaccurate cadreing which incurs because of the fact that finder and objective are not in the same level. Instead of this type of finder, use of LCD is recommended if possible.
LCD finder on compact cameras - displays in real time what objective sees. This is possible because of camera construction where light always falls on sensor. Good way of cadreing because of parallax absence and almost ideal cadreing. Newer camera generations have shiftable LCD finders, that even more increases their usableness. Difference between LCD display on compact and DSLR cameras lies in fact that compact have constant scene display, which is with DSLR's still mostly impossible because of the work mechanism.
Optic finder on SLR cameras - this finder system enables us to see the scene objective projects on sensor. Transffer of image is performed by system of mirrors and prism, and it is immune to parallax phenomenon. The way this system works is that before trigger pressing, mirror is in lower position and covers the sensor and projects scene in finder. In the moment trigger is pressed all the way, mirror lifts and sensor is exposed. In this time, there is no display in finder because mirror is lifted up to enable light arrival on sensor. After exposure time has expired, everything returns to previous position, and we have scene display in finder. This is the main reason we still don't have "live preview" on SLR cameras LCD, but instead it serves us to afterwards view taken photographs.
Digital finder/electronic view finder (EVF) on compact digital cameras - basically the same system as LCD finder on compact digital cameras, just smaller dimensions and positioned above big LCD. Its purpose is saving batteries, and has ability of better displaying when photographing on intense light, when outer LCD is not the best solution.