True wood veneer consists of very thin sheets of real wood shaved off of logs. It can look like the grain of plywood if it is peeled off of a rolling log, or look like furniture lumber if it is sliced flat like boards right across the log. It was developed as a way to use sturdy but inexpensive boards as the core of a board or large surface while having expensive exotic species of wood on the apparent surface. These thin strips were also the easiest way to create the intricate inlay work that you see on some fine furniture. Modern plastic laminates are direct descendants of real wood veneers.
Working with this thin and rather delicate wood requires special very sharp tools.
Veneer snips look like scissors but they have small blades for ease of movement and are extremely sharp, and often serrated so that the wood will not slip.
Sharp mat cutters are often used for cutting veneers, providing a good way to cut wide curves.
Veneer saws have curved blades to allow you to cut without starting at the edge.
Special parallel knives permit cutting strips, straight or curved, that have perfectly parallel sides.
One of the tricks used when creating a veneer inlay of different coloured woods, is to tape one veneer over another and cut through both at the same time with very sharp very thin knives. When both cut pieces are removed, one will fit perfectly inside the other.
Veneer usually has a bit of natural warp, since it is a real piece of wood. The glue will soften up the wood and allow it to be forced flat. If you are using regular wood glue, it is necessary to press the veneer tightly while the glue sets. If you are using contact cement, you will want to have a veneer roller. Rather than scraping over this delicate and probably expensive veneer, the roller rolls over it to force it into the contact cement and down flat to the base material.
Follow this link for tips on successfully applying veneer to the edges of presswood and plywood.