Proofs are an essential part of printing. The proofs that you can expect depend on what's being printed.
If it's being machine-set I send out galley proofs and page proofs for checking and approval. Galley proofs show the type as it comes off the caster, and page proofs show it made up into pages. They are generally proofed in black only on proofing paper. Small pieces of setting are scanned and sent as PDFs.
A smaller item, like a letterhead or business card, is usually sent as page proofs.
When I'm having plates made from digital artwork I send PDFs of the colour separations.
Sometimes customers want to see machine proofs, printed in colour on the correct stock. I have to do all the work of setting up the machine in order to provide them, so it can be expensive. Machine proofs of a short-run job can cost nearly as much again as the job itself.