An LLC is run either by its members directly or by one or more managers the members appoint. The choice goes in the operating agreement, and it decides who can sign contracts, open accounts and commit the company. With one owner it hardly matters. With several it matters a great deal, and it's one of the first things a bank checks.
Member-managed
The members run the company and make decisions together, by a majority or however the agreement says. It's the default in most states when the agreement is silent, and it's how nearly every single-member LLC works: the owner manages, signs and decides. Two or three co-founders who all work in the business usually choose this, with the agreement listing the decisions that need everyone's consent.
Manager-managed
The members appoint one or more managers, who don't have to be members, and the managers run the company. The members' role is to appoint and remove managers and to vote on the major decisions the agreement reserves for them. It suits a company with passive owners, such as an investor or a relative who holds a share but doesn't work in the business, or one where a single founder runs things with the others' agreement. It also suits a holding company run by one person for several owners.
What it changes
Who can sign, in Wyoming and Delaware
The two states differ on who can commit the company by default. In a member-managed Delaware LLC, each member can bind the company unless the agreement says otherwise. In Wyoming, a member isn't the company's agent just because they're a member, so the operating agreement should say plainly who can sign. Either way, write it down: the bank will ask.
What's on the public record
Neither Wyoming nor Delaware asks for the management structure on the formation document, and neither lists managers publicly. It lives in the operating agreement, which the bank reads to see who can open the account and sign. A manager who isn't a member is still identified to the bank as a person who controls the company, even without owning a share.
Changing it
Amend the operating agreement with the members' consent, as the agreement requires, and tell the bank. A member-managed company often switches to manager-managed when it takes on an investor, with the founder as manager.