Thompkins was the passenger and shooter in a drive-by in Michigan, hitting two victims. One of his victims died, the other survived (and later testified against him, identifying him as the shooter). Thompkins stripped and abandoned the van and fled the state. He was apprehended in Ohio, and two Michigan detectives interrogated him there.
At the beginning of Thompkins' interrogation, he was advised of his Miranda rights. He was given a form with the standard warnings, he read one of the warnings out loud and a detective read the rest of them. He refused to sign the form, and there was conflicting testimony as to whether or not he verbally acknowledged that he understood his rights. Thompkins was then interrogated for about three hours. He didn't speak much at all during the entire interrogation, but in the last fifteen minutes one of the detectives asked him if he prays to God for forgiveness for shooting that boy down. Thompkins answered "yeah."
Thompkins was later convicted of first degree murder (among other things). Evidence of his confession was used against him at trial, as was the testimony of the driver (who had been convicted of violations of some firearms laws, but acquitted of the murder). Thompkins appealed his conviction. He argued that his confession (such as it was) should have been suppressed because by remaining silent during the interview he had invoked his Miranda rights, or alternately that the police should not have interrogated him because he did not waive his Miranda rights (there's a subtle distinction there). He also argued ineffective assistance of counsel based on a minor flaw with the jury instructions which his lawyers didn't catch (the Supreme Court held that this lacked merit because there was plenty of evidence with which to convict him).
The US Supreme Court upheld Thompkins' conviction, and clarified some issues surrounding Miranda advisements.
The Court had previously ruled that when a suspect invokes his Miranda rights, the police must stop the interrogation. Previous decisions had also held that when an invocation of the right to counsel has to be made unequivocally. If a suspect makes an ambiguous or unclear reference to this right, then the police are not required to stop the interrogation or to clarify the suspects intent. In this decision, the court extended that same reasoning to the right to remain silent; in order for a suspect to stop an interrogation he must clearly state that he wants a lawyer or that he is invoking his right to remain silent. Simply remaining silent does not end the interrogation.
On the other hand, simply remaining silent is also not an effective waiver of one's rights. In order for a waiver of one's rights to be valid, it must be voluntarily (meaning "the product of a free and deliberate choice rather than intimidation, coercion, or deception"), and the waiver must be made "with a full awareness of both the nature of the right being abandoned and the consequences of the decision to abandon it." But there is no requirement that the waiver be expressed, a waiver of ones rights can be implied by ones actions. The court held that there was no evidence of coercion during this interview (there was nothing inherently coercive about the three hour length of the interrogation, no other coercive tactics employed, and Thompkins wasn't in fear of the detectives. The detective's reference to God also did not render the confession involuntary; the court pointed out that "the Fifth Amendment privilege is not concerned 'with moral and psychological pressures to confess emanating from sources other than official coercion."). The court also held that there was sufficient evidence that Thompkins understood his rights (and his understanding of his rights was never disputed at trial), and that by answering some of the detectives' questions (even if it was after almost three hours), he engaged in a course of conduct which implied a waiver of his rights. All of this was enough to satisfy the requirements of Miranda.