I am part owner of a small business in Texas. I'm trying to figure out how to pay myself a salary. Can somebody please explain how this works and the associated tax considerations?

Let me see how do you own the business, partnership, LLC incorporated ?

You basically write yourself a check, depending on how you formed the company is the tax variations. Not meaning this wrong, but if you don't know how to pay yourself, you need serious accounting help, pay a local accountant to make sure you are set up properly.