The rules in Santa Clara county about open burning are simple. Only two kinds of burning are allowed: recreational fires and agricultural burns.
Recreational fires - used for warming, cooking, religious, or ceremonial purposes - are allowed within limits. They must be small - less than three feet in diameter - monitored by a responsible adult at all times, and equipment to extinguish the fire must be present. If the burn is in a fire pit - contained by a rock ring for example, like camp fire - it must be at least 50 feet from any structure or combustible material. If the fire is in an approved container - an outdoor fireplace or similar item - it may be closer, but still must be at least 15 feet from any structure.
Agricultural burns are more tightly controlled. To qualify for an agricultural burn permit you must derive income from the agricultural materials being burned, so an orchard or vineyard would qualify if you're selling the fruit, but not if it's just for personal use. And it must be a commercial scale operation; a few fruit trees won't qualify. Burn permits are issued at Redwood Estates fire station, usually for just one day to accomplish a single burn. No inspection is required, but you must get a permit from the station in person before burning.
In addition, it must be a permissive burn day as determined by the Bay Area Air Quality Management District. Call 800-792-0787 and listen to the recorded message to determine the burn day status. Burning on no-burn days is prohibited.
Note that you can be held responsible for any fire that escapes and causes injury or property damage. In such a case you can be liable for the costs of suppressing the fire, the property damage, and treatment of injuries suffered. Also note that if anyone complains about your fire - even a recreational fire - because of the smoke, smell, or discomfort over possible spread, it may be classed as a nuisance fire and is then illegal. You will be instructed to extinguish it in such a case. In essence, if you have a burn of any kind it needs to be acceptable to everyone affected by it.
To be clear, the backyard burning of brush & tree trimmings (or anything else) is not allowed in Santa Clara county. Hazard reduction burns aren't legal either. If you've trimmed your trees or cleared brush you'll have to chip the trimmings on site, compost them, or haul them to the dump to dispose of them.
This article written with the generous help of Captain Matt Brown of Santa Clara County Fire, who was regularly stationed at the Redwood Estates fire station at the time the research was done.