A common feature of all liability insurance policies is that the insurer has a contractual duty to defend the insured any time the insured is sued for something that is potentially covered under the policy. This means that the insurance company will hire an attorney to file an answer to the lawsuit and protect the insured's interests. This is a valuable right because the cost to defend even a small or seemingly frivolous lawsuit can be significant. Although the insurance company pays the defense attorney's fees, Texas law is quite clear that the attorney's client, and the one to whom a fiduciary duty is owed, is the insured.
All liability policies place a duty on the insured to notify the insurance company if and when the insured gets served with suit papers. Any such papers should be provided to the insurance company immediately. This is known as "tendering the defense" to the insurer and is what triggers the duty to defend. The failure to notify the insurer promptly can cause the insured to lose coverage if the insurance company is prejudiced by the delay.
In Texas, the insurer's duty to defend is determined by the "8-corners" test. This means that the four corners of the petition filed against the insured are compared with the four corners of the policy. If there is anything alleged in the petition that is potentially covered by the policy, the insurer has the duty to defend the entire case. An important aspect of this test is that the duty to defend exists even if it is known that the allegations are false. For example, if an accident is alleged to have occurred on a date in which a policy was in force, the insurer will have a duty to defend the case even if it is known that the accident actually happened after the policy had expired.
In a situation where the insurance company has a good faith basis for believing that a claim is not actually covered under the policy under the true facts, it may choose to "reserve its rights" to deny coverage. This means that even though it is defending a case, it may refuse to settle the case or pay any resulting judgment. Generally speaking, if you are being defended by an insurance company under a reservation of rights, you should consult with an attorney with regard to the practical effect that this may have on your situation. (Note that your defense counsel will typically not be willing to provide assistance in this regard, so you may need to retain your own "coverage counsel". )
In some situations, the insurance company may even sue its own insured in a declaratory judgment action in order to seek a judicial determination of whether coverage exists. If this occurs, it is imperative that the insured retain competent counsel to defend the action.