I used to work for County Government in California and I was the Property Tax Accountant, so I hope you find this helpful.
The annual form you fill out every year is called a Business Property Statement and it lists all the property you own, possess, or control in the conduct of a business, profession, or trade. On this form, you list the year you acquired the property and the purchase amount. This information gets entered into a database (in this case at the County level) and those assets get assigned a class code for depreciation purposes. This is the basis for how the book value of each asset will be determined for each tax year.
The penalty someone mentioned is for late filing the statement. Typically, in CA it's 10% of the gross assessment. So, for example, someone who has $10,000 in business property can be penalized $1,000 for filing a late business property statement, increasing their total assessed amount to $11,000. Keep in mind, most tax rates are generally somewhere between 1 and 1.2%.
Unsecured property tax is for items that can be relocated and are not real estate. Hence, why it's called "unsecured" versus "secured" property taxes (property tax you pay for your residence).
Examples of items taxed on the Unsecured property tax roll include business property, aircraft, vessels, boats, and personal watercraft.
One caveat: The Assessor is required to assess all property owned/based in the county as of Jan 1st - called the "lien date". So, if you purchased an item anytime after Jan 1st, it escapes taxation for that year and the first year it will be taxed is the following year.