The Federation is a representative republic, with an elected president as the head of the entire interstellar state. An election is held every four years, and a president may serve for an unlimited number of terms.

Political and direct administrative power is held within the Federation Council, which is composed of one councillor from every Member World. There is no limit as to how many terms a person may serve as councillor. T'Latrek of Vulcan, for instance, served on the Federation Council for nearly a century. Each individual Member determines how its councillors will be determined; the First Minister of Bajor, for instance, nominates that world's councillor and the Chamber of Ministers ratifies him or her, while the electorates of many other Members elect their councillors directly.

The Federation government has several executive departments whose heads form the Presidential Cabinet, who advise the president on their issues of jurisdiction and run their departments on a day-to-day basis. Cabinet members can have strong influence on Federation policy based upon their work with the president and the appropriate members of the Federation Council.

By the late 23rd and 24th century, the capital city of the Federation is Paris, and the capital planet is Earth. The seat of government is the Palais de la Concorde.

• Memory Beta article on The Federation

A bunch of people set up public bulletin boards, and agree to copy whatever gets posted on one of them to all the others.

That is a very nice photo.

The Wikipedia entry does a very good job in its first few paragraphs:

The fediverse (a portmanteau of "federation" and "universe") is an ensemble of social networks which can communicate with each other, while remaining independent platforms. Users on different social networks and websites can send and receive updates from others across the network.

ActivityPub, a W3C standard, is the most widely used protocol that powers the fediverse. Noted fediverse platforms include Mastodon, Lemmy, PeerTube, Pixelfed, and Threads.

The Terran Federation is the government of Earth and its Space Colonies. It is a multicultural society that functions as a limited democracy. Full citizenship is not guaranteed to all members of the population and must be earned, typically through military service. Citizenship grants additional civil rights, such as suffrage and the right to run for political office.

People of higher levels of authority also have to suffer tougher repercussions of their actions. For instance, a Lieutenant could hang for making a mistake that a Private would merely be dismissed and maybe lashed for, while Corporal and Capital Punishment are practiced by the government as well as physical child rearing being standard use amongst the population.

Everyone is born a "Civilian", and at age 18 every "Civilian" has the right to enroll for a minimal 2 year term of "Federal Service". In theory a completed term of Federal Service ensures a "Citizen" is willing to put the needs of the community before their own personal well-being. This is because Federal Service is tough and dangerous (by design). It can involve joining the Military, being a Human Guinea Pig, testing survival equipment or Manual Labour.

The Federation makes it quite easy to quit a term of service before completion (even during war time), but once someone has quit they are never allowed to enroll again. This is to ensure that all volunteers are dedicated, whilst also discouraging people from leaving.

"Civilians" are neither discriminated against, nor deprived of legal rights other than that of the ballot.

Once in the military a volunteer has the choice to "go career", choosing to devote 20 years of service to the Federation instead of the usual 2 years required to gain Citizenship. After these 20 years they can then leave and get a "reserved job", for example in the police.

• Starship Troopers Wiki article on the Terran Federation

Possibly not for a five-year-old, but most people have an email account, and technically this is a federated network. Tuta, Proton, Posteo, and all the others are independent networks, but they can communicate with each other (unlike Facebook, Twitter, and others which require an account on each platform for communication).

Forum: People set up communities on their or someone else’s server. People have to individually find and register for each siloed community.

Aggregator site (i.e. reddit): Everyone sets up communities on the same server. A single account on the server can join and interact with any community they want, and aggregate them all together into a feed.

Federated aggregator (i.e. Lemmy): People set up communities on their or someone else’s server. They all use the same language so they can communicate with each other. You can make an account on any server and then visit any community, and aggregate them all together into a feed.

Simple English Wikipedia might help.

https://simple.wikipedia.org/wiki/Fediverse

Imagine you have multiple groups of people sitting in circles passing notes between each other. Optionally those notes can be copied to other groups.