If you found that your luggage is damaged when you received your checked luggage from the baggage claim area, present the damaged luggage and the claim checks to the airline baggage office.
Point out the damage to the officer on duty, and make sure you have your claim checks available. Most airlines have up to 48 hours limit for declaring damages.
(Please take note that for some budget flight, no claims will be entitled if you did not purchase travel insurance for your flight)
The airlines are selective about what type of damage is covered by its liability policy. What everyday people consider 'damage' is considered 'normal wear and tear' by the airlines. Items that are considered 'normal wear and tear' for many airlines includes wheels, straps, handles, zippers, pockets, name tags, and most things that protrude from your luggage as well as minor tears, cuts, or soiled. This may vary between airlines.
If your damage claim is accepted, you may be offered one or more settlement possibilities, to include things such as bag replacement, repair, or in cash.
In general airlines regard the purpose of luggage to be the protection of its contents during transit. If the luggage is damaged, even severely, but the contents remain unharmed then airlines regard the luggage as having fulfilled its purpose and will not compensate owners.