Finches are a type of bird that ranges in size from the 2-inch Siskins to the 9-inch Grosbeaks. Most birdhouses built for finches will be built to accommodate the House finch, a small, seed-eating finch that is common throughout North America. The House finch prefers a 2-inch opening in a house raised off the ground, out of the reach of predators.
The House sparrow is an introduced species that has reduced House finch numbers throughout much of its range by taking nest sites and even destroying eggs and young. House sparrows require small perches in order to gain entrance to a birdhouse, so cutting your birdhouse opening and neglecting to install or removing the perch will allow the House finch to gain access without admitting sparrows.
If you construct your birdhouse out of hardwood, such as oak or maple, consider using a softer wood for the front panel, where the opening will be. This enables a House finch to widen the opening if need be. If you use a soft wood, such as pine, for your front panel, make your opening slightly smaller -- in the vicinity of 1 3/4 inches in diameter should be sufficient. If you use hardwood for the front panel, make the opening the full 2 inches, as the finch will not be able to easily widen the opening.
Trial and Error
Successfully housing a specific species of bird can be difficult and may take several seasons. Many birds have similar needs when it comes to openings and house locations, so you may find you are housing a family of chickadees, rather than House finch. Tweaking with the hole opening from season to season can help you zero in on your target species. A hole that is too large can easily be remedied by wood-gluing a small square piece of wood over the hole in the off season and cutting a new hole through the square with the desired diameter.