Public projects almost always require bid bonds; private projects sometimes. Bond amount is typically 5-10% of the bid value. Submitted as a bond from a licensed surety, a cashier's check, or a certified check.
Don't confuse bid bonds with performance bonds (which guarantee project completion) or payment bonds (which guarantee subcontractors and suppliers get paid). All three are typical on public projects; the bid bond is the prerequisite for being eligible to bid.
Bond cost to the contractor is typically 0.5-2% of the contract value depending on surety relationship and bonding capacity. Build the bond cost into your bid as a separate line or roll into general conditions.