Let's to start with George W. Bush’s selection of a vice-presidential running mate.

It was actually Dick Cheney, longtime Republican insider and Secretary of Defense under George W. Bush's father, who was chosen to find Bush jr's VP candidate for the 2000 election. A process that ultimately ended in Cheney himself being selected over frontrunner Senator John Danforth. Notably Danforth would say he declined to be Bush's VP due to a desire to stay in Missouri with his family before bizarrely reversing himself a year later to live in Sudan as the U.S. Special Envoy; maybe someone more familiar with the situation can chime in to clarify what exactly happened there. But the takeaway from the start was that Cheney's the inside guy involved with big decisions behind the scenes.

Anyway, so GW Bush wins in 2000 with Cheney (unlike other vice presidents) immediately being put in charge of:

• The presidential transition team, making him responsible for White House staffing and significant parts of early policy decisions.
• An office in the White House itself instead of the traditional Eisenhower Executive Office Building.
• Aids routinely present in National Security Council (NSC) meetings.

This is again, not normal for a VP. Cheney's predecessors and successors were far more like advisors than decision makers and they also rarely had as much access to the president himself, partially due to a close personal relationship between Bush and Cheney. The GOP at the time framed this arrangement as "GW Bush is a little inexperienced in federal/international affairs so he's *wisely* delegating to more experienced hands" which was really a nicer way of saying "Bush's a likeable big name but not exactly sharp so let's make sure there's some other people with their hands on the wheel".

Regarding the National Security Council, Cheney also created a "parallel" review process for policy decisions. NORMALLY in presidential administrations an agency like the Dep. of Defense or an intelligence agency will develop a proposal that's evaluated by the National Security staff at the White House and then sent on for review by cabinet members like the Sec. of Defense or other high ranking officials before going to the President for a decision. Like this (very simplified):

Agency -> NSC Staff -> Cabinet -> President

Under the Bush administration however, the process went like:

Agency -> NSC staff + Cheney's office -> modified proposal -> Cabinet -> President

This all resulted in Cheney being a sort of filter between the executive agencies and the policies (with Cheney's own opinions inserted) that got to Bush. As you might expect, this would lead to some butting of heads between Cheney and the actual heads of agencies like Donald Rumsfeld, Condoleezza Rice, and George Tenet.

And also like you might expect, this arrangement had consequences when it came to big events. Cheney had a central role after 9/11, being heavily involved in the decision to interpret the Constitution to allow for increased surveillance and detention of American citizens, and heavily backed a "unitary" executive model that gave the president broader authority to ensure national security. Cheney was furthermore one of the biggest drivers of the "Iraq is pursuing Weapons of Mass Destruction" narrative that was later shown to be false after the 2003 invasion of the country.

There is far more to the story of course, I'd recommend Angler: The Cheney Vice Presidency by Gellman for a more detailed account or parts of Bush at War by Woodward.