Yes, you've got it right.
Under the old system, you had a standard deduction plus a personal exemption. The standard deduction for a single adult in 2017 was $6350, and the personal exemption was $4050 per person in your house. So if you are single with no kids and a $15k income, in 2017 your taxable income was $4600. In 2018, the standard deduction for a single went up to $12,000, but the personal exemption went away, so your taxable income in 2018 would be $3000.
In addition to the change with the deduction, the tax brackets were adjusted, meaning the tax rate paid on the taxable income is lower for most income levels. However, at the level we are talking about, the tax rate remains a flat 10% in both 2017 and 2018. As a result, in your scenario, the total tax burden would be $460 in 2017, and $300 in 2018.