Who pays unemployment taxes?

In Texas, Your employer ultimately pays 100% of any unemployment benefits you receive.

The state issues your check from its account. So in that sense it appears that the state pays you the benefit check. But then they bill the employer for the amount of those disbursements. So in actuality, the employer is still paying it. In fact, Unemployment Commission employees here will not even call it "Unemployment Insurance" because it is in effect not insurance in the way it is handled here. They use the term "Unemployment Compensation" instead, or at least when talking to the employer.

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The employer pays into a state fund (SUI) and a federal fund (FUTA). Below is a link explaining how it works in Arizona. It generally works the same way in other states.