The above is a very good answer. I just want to add/clarify a couple of points.
For an employee-owned vehicle, the corporation can reimburse the employee for business use of a personal vehicle. The reimbursement is always a business expense for the corporation, but it is only tax-free to the employee if the business has an accountable expense plan. This is explained in chapter 4 of publication 463. https://www.irs.gov/pub/irs-pdf/p463.pdf.
If the business does not have an accountable plan, then the employee reimbursement is considered a form of wages and must be included as taxable wages on the employee's W-2 and is subject to employment tax and income tax withholding.
For a company-owned vehicle, the company can deduct ordinary and necessary business use. However, if the employee is allowed to take the company vehicle home and use it for personal use also, the value of that personal use must be included in the employee's W-2 taxable income. There are a couple of ways to calculate this. (And there is an exception if the vehicle is so distinctive and specialized that it can't reasonably used for personal use. For example, a school bus driver who drives the bus home so they will have it first thing in the morning, will not be taxed on personal use even if they stop at the grocery store on the way home once in a while.)
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