Other than seeming like a member of some secret organization the UK elects shadow members. What are they? Is there any corresponding member of US government?
The Shadow Cabinet consists of senior members of the Opposition (the largest party not forming part of the government). Each of these have a brief to represent a direct opposition ("a shadow") to the corresponding member of the government cabinet. In principle this means that alternatives to government policy are developed and expressed by a politician who has knowledge and experience in the area, and who could immediately take over the role in the event of a sudden change of government.
The idea is relatively recent, dating back only to the 1950s in the UK. As far as I know, there is no kind of corresponding organisation in the US, since there the executive cabinet is not formed from sitting legislative politicians and there is far less potential for an unforeseen change in the party in power. Until 2011 the Prime Minister had the de facto ability to dissolve parliament and call a general election at almost any time. Even now, a government which proved unable to govern could trigger a dissolution.
Supplemental to the other answer, the shadow cabinet is both a government-in-waiting and a day-to-day means of providing opposition to ministers on an issue-by-issue basis. Just as in football the defenders are there to 'mark' attacker.
Why the need for a government-in-waiting? Until 2011 Parliament itself could change the PM and hence the Cabinet at any time on a confidence vote with no need for elections. I'm not sure if this is still true after the Fixed Term Parliaments Act.
Theoretically, the PM could be replaced at any time by the Queen. It has been some time since this power has been used in the UK, but it was used in 1975 in Australia.
It's also there in the case of dire emergencies or disasters. Had the 1985 IRA bombing of the hotel holding the Conservative party conference been bigger, it could have killed the PM, Cabinet, and enough MPs to lose the majority in the House of Commons.