Anyone who wishes to breed their finches knows they will need to invest in a nest and/or nesting material. Nests are available in many shapes, sizes, and materials these days. Just as nests have diversified so has the nesting material. We can give our finches grass, burlap, coco fiber, nest pads, tissue paper, feathers, and so on to use as nesting material.
So, what works best? All of the above. If you buy your finches from a local breeder, ask him or her what types of nests these finches were bred in. They will be more comfortable in the same types of nests they hatched in and may breed more readily.
If the breeder can not be contacted it's best to buy the type of nest which will fit your cage best. The nice thing about finches is most will nest in just about anything as long as they feel safe and secure. I have always used covered bamboo nests, both the regular and large sizes depending on the species and history of each pair's average clutch size.
Species like the European Goldfinch will prefer the open 'Canary' style nest and will probably never breed in any other type of nest. Others like Gouldians are often bred in the larger nest boxes, which hang partly outside the cage leaving more space in the cage available for exercise. Each type of nests has its good and bad points.
Canary Nests: These are bowl shaped nests. They work great for finches with longer bodies and tails. Any finch with a Canary size and shaped body will probably prefer one of these nests, though other species will use them as well. These are available in bamboo, grass, millet, and plastic. While they make checking the eggs and accessing the young easy, they aren't as secure as the covered nests which is why many species prefer covered nests. These nests can be cleaned and used a few times before they are pitched (with exception to the plastic ones which rarely go bad). They are also very inexpensive.
Finch Nests: These nests are more oval shaped with a covered top and hole in the top front section. They come in 2 sizes, regular and large. Zebras, Society, Nuns, Grass finches, Owls all use these types of nests as well as most any other style of nests. These nests are available in bamboo, grass, and millet. While they offer a more secure feeling for the finches they do make it much more difficult for us to check eggs and young. They also take up space inside the cage. These nests can be cleaned and used a few times before they are pitched (with exception to the plastic ones).
Nest Boxes: Literally they are wooden box nests. Most have a perch just outside the hole and a small door on the top so the breeder can check on the eggs and young without disturbing the parents much at all. These are mainly designed to be hung on the outside of the breeding cage with only the nest box opening inside or flush with the wall of the cage. I've only seen these available in wood, but I'm pretty sure there are plastic ones available somewhere. These nests can be used many times over if they are cleaned and taken care of when not in use. These are considerably more expensive than the others.
Nesting Material is the next big step with many options. You'll want to pick a material that it suited to your finch. For most finches that means small and light in weight yet soft enough to line the nest. My personal favorites are grass and tissue paper.
Grass: Grass is cheap, especially if it's growing in your yard, and bio degrades very quickly. Grass should be dry and parasite free before giving it to your finches.
Coco Fiber: Coco Fiber is also cheap and bio degrades very quickly. Coco Fiber is favored by weavers and most Mannikins, yet most all grassfinches enjoy it also.
Burlap: Another cheap way to go. You can buy burlap at most all craft or fabric stores and it's very inexpensive. You'll want to buy the loose weave, as its easiest to pull apart. I cut the burlap into 2 or 3 inch squares and pull them apart. The finches can then collect the strands and work on the nest.
There are some materials which shouldn't be used, such as; newspaper, cotton, some wood shavings or string. Newspaper ink and some wood shavings can be toxic to finches. Cotton, string, even some yarn can become entangled around your finches leg or other extremities and lead to amputation or death.
Now please note, just because your finches have a nest and material doesn't mean they are going to build it or raise a family. They must feel secure in their home environment and be healthy before all else. I will also point out that if you don't want your finches to ever breed, don't give them a nest.
It has become a very common misconception that finches need a nest to sleep in. I have no idea where this started but it's spread worldwide. I totally agree that finches do look very cute all snuggled down in their nests. Many species will cram as many individuals in a nest as possible, it's adorable seeing 6+ heads looking back out at you through the opening. But; just because it's cute, doesn't mean it's correct.
In the wild, finches build nests to breed; that is their purpose. The nest is worked on for a few weeks maybe longer, the female lays her eggs, incubation leads to hatching. Finally after about 2 months the nest is abandoned and the family moves on. Some species will nest twice in a year but most do not. The rest of the time they sleep on branches in the trees and shrubs with no complaint.
Finches in captivity have all the same instincts as their wild counterparts. When given a nest, they will claim it and wait for a female to come a long. If a female is present they will make all attempts to breed. Granted even without a nest, attempts to procreate will be made, it's a very powerful drive in all living things after all.
The best method to stopping breeding is to simply remove the nest. Without a nest, and without the means to build a nest production of offspring is very hard or impossible. No nest = No babies. This doesn't mean the finches will stop mating or laying eggs. These activities are normal and will continue even in same sex flights.