If you don't buy insurance through the rental car company, and have no coverage through travel insurance, credit cards, or your home auto insurance, then the only liability insurance you will have is the minimum coverage the rental car company must provide in most states.
The minimum liability insurance you must have, by law, in Texas is detailed here, along with other helpful information about car insurance in the US:
The current minimum liability limits are $30,000 for each injured
person, up to a total of $60,000 per accident, and $25,000 for
property damage per accident. This basic coverage is called 30/60/25
coverage.
This describes liability insurance, which is mandatory insurance that covers you for damages you may cause to other people or property while driving. Note that this limit is very low. It is quite possible to do far more damage than this while driving, and you may be sued for the difference. Remember that medical costs in the US are expensive; a car accident victim can easily run up a hospital bill far more than $30,000. In the unfortunate event of a fatal accident or one that leads to serious long-term disability where you are found to be at-fault, you could incur significant liability. It is not uncommon for rental car companies, such as Hertz, to offer $1,000,000 in supplemental liability insurance.
The rental car company, or a third-party, will offer to sell you supplemental liability insurance, which will increase these limits and allow you to better protect yourself.
Separately, there's insurance for the car itself. You will not have any such insurance unless you purchase some, whether from the rental car company (CDW or LDW, which isn't strictly insurance, but is the rental car company agreeing you're not responsible for damage to their car) or a third-party. If the car is damaged or destroyed, stolen while in your possession, filled with flood water, dented in a hailstorm, or spontaneously combusts, whether your fault or not, you are responsible for paying for it. You might be able to recover costs from someone else's insurance if they hit you and they're at fault (assuming they have insurance), but the rental car company will come after you to pay up. They'll also charge you for the loss in revenue they incur while the car is being repaired. Unless you're prepared to pay for the car on the spot, insurance is a good idea.
In addition, the rental car company may offer accidental medical and/or death insurance to cover you and your passengers. This is likely already handled by your travel insurance and may be unnecessary. They also may offer Personal Effects Coverage, which will pay you if your property is stolen from the car. Again, this may already be handled under your travel insurance and is best avoided by not leaving property unattended in your car.
The New York Times has a brief guide to the options that may prove useful.
