Most courts are reluctant to remove the executor of an estate --- the deceased wanted her to oversee probate, and judges need a good reason to go against the wishes of the person who has passed away. If you think you have grounds, you can petition the court for removal of the person, but you must have an active interest in the estate in order to do so. You must either stand to inherit or the estate owes you a debt. The executor has the right to tell the court her side of the story, however, and to try to keep the job.
The easiest way to remove an executor is if, in the time that has passed since the deceased wrote his will, something has occurred to disqualify the person he named. For example, she may have committed a crime. Most state laws do not allow anyone with a criminal record to serve as executor. Or perhaps she suffered a disabling medical event, such as a stroke, and is no longer up to the job, either physically or mentally.
Failure to Perform
If the executor accepts the job, then does nothing toward probating the estate, you can file a petition based on her failure to perform. In fact, in most states, the court will automatically move to terminate her position after a certain amount of time has passed without her taking any action on behalf of the estate.
Conflict of Interest
If you think the executor cannot perform his duties impartially for some reason, you can bring this to the attention of the court. Ultimately, however, it comes down to the opinion of the judge whether the conflict carries such weight that it impedes the executor from fairly representing the estate. For example, if the executor owns stock in some asset that needs to be liquidated to pay estate debts, and if he refuses to sell the asset until he can get top dollar for it, this might be a conflict of interest. He would be acting in his own best interests, trying to get the highest return on his personal investment.
Misconduct must generally be egregious to get an executor removed. For example, the executor would have to refuse to do as the deceased specifically asked in her will, fail to obey a court order, or fail to protect assets, resulting in them being lost or damaged.
Certain behaviour can be distressing to beneficiaries, but not constitute grounds to have the executor removed. For example, you can't file a petition because you think he's not pushing the will through probate fast enough or because he is hard to get on the phone. You can't have him removed because you saw him having a martini at lunch, although if he receives a DUI conviction, you would probably have a case. Other things that don't constitute grounds are if he is rude to you or invests the estates assets in a way you disagree with.