The Minister was an unseen man from the Ministry who presumably founded the Anomaly Research Centre. He had overall control over the ARC operation but he appointed James Lester as head of operations. Lester always kept the Minister fully informed on all operation decisions (except in some cases).
It was presumed "the Minister" was actually the "Prime Minister of the United Kingdom"; if he wasn't, then he probably was the "Minister of Defence" or "Minister of the Home Office". Going by the real world, it was likely that the Prime Minister in Series 2-3 was Gordon Brown, and the Prime Minister in Series 4-5 was David Cameron.
When Tom Samuels and James Lester worked in planning civil contingencies, Tom went over Lester's head and presented a plan to the Minister, however he asked for Lester response so the latter pointed out all the weakness in the plan which resulted in the Minister firing Samuels. (Fire and Water)
After the second Forest of Dean expedition, the Minister sent a top priority memo to Oliver Leek, suggesting that a Public Relations manager should be hired to assist the ARC operation because he thought things with the media were "getting out of hand". When Leek told Lester, the latter initially soured to the idea because he didn't think one was needed but almost immediately changed his mind when he needed to cover up Raptors in a Shopping Mall and so hired the Minister's suggestion: Jenny Lewis.
Charles Bairstow attempted to persuade Lester to mount an expedition to Peru to rescue his sons, but Lester refused. Charles contacted the Minister and convinced him to make Lester order the expedition thus sending the main ARC team overseas.
After Nick Cutter's untimely demise, Christine suggested several candidates to the Minister for replacing Cutter as the ARC's team leader, all of them were of a military background.
Lester got the Minister's permission to appoint Danny Quinn as the new team leader.
When Christine alerted the Minister to the ARC's possession of the Artifact, he demoted Lester and appointed Christine to become the head of the ARC operation. Later, Christine described the Minister as "a Whitehall glovepuppet with the IQ of a root vegetable". Becker recorded her saying that and sent it to Lester who forwarded it to the Minister. He then reinstated Lester as the head of operations, and Christine was forced to leave for a meeting with the Minister, who was said to be very upset.
The Minister made the decision to suspend the Anomaly Research Centre after the disappearances of Danny Quinn, Connor Temple and Abby Maitland and the death of Sarah Page. Lester suspected he would have good cause to regret this decision after the appearance of a Stegosaurus in the member's bar in the House of Commons. (Webisode 1)
Approximately four months later, the ARC was revived with a public-private partnership between Prospero Industries and the government, meaning Lester had to keep both the Minister and investor Philip Burton "happy". (Webisode 1, Webisode 2, Episode 4.1)
The Minister became good friends with Philip. At one point, the two of them had dinner together and the Minister persuaded Philip to invest another half-billion pounds into the ARC operation. Lester became bitter about Phillip's friendship with the Minister so he asked Jess Parker to keep an eye on the Minister's calendar, however, she refused.
Lester told Matt that the Minister was involving other government organizations to catch Ethan, but Matt convinced him to give himself another day to find the criminal and look though the evidence they had collected and joked that he would get Lester back on the Minister's Christmas card list.
Upon meeting Lester for a joint ARC-Navy operation, Admiral Marston revealed that the Minister had briefed him on the ARC operation, six months prior.
After the New Dawn Anomaly closed, Lester got a phone call from the Minister. He informed Lester that a train had gone missing from King's Cross Station, and that an Anomaly may be there. At that time, the Anomaly Detection Device was still without power.
While at the Home Office, Lester mentioned that he reported to "the Mayor" who could have been the Minister in the original timeline.
Lester received a call from the "Prime Minister" and put it through to Admiral Marston. It is unknown whether it was "the Minister" or not.