One account per person

Each person plays with one account. Having more than one is strictly forbidden, even if you never use both at the same time and even if one never helps the other. Two accounts belonging to the same person is what we call multi-accounting, and multi-accounting gets a permanent ban.

Anyone caught with more than one account loses all of them, not just the newest one.

What the game allows, and encourages, is having several characters on the same account: up to four, one playing and the others hunting offline. That is how you get more than one trainer without breaking any rule. See Trainer License.

Playing from the same house is normal

This was never a problem and still is not. Siblings, a couple, friends sharing the same internet: several accounts coming from the same place is common, and nobody is punished for that alone.

The rule is about one person running several accounts to benefit themselves, not about a household with more than one player.

What is not allowed

Opening a new account for yourself. That is the main rule.
Referring yourself. Creating an account and using it as your referred friend is blocked. See Founders and referral.
Multiplying a per-person reward. A reward meant for one player cannot be claimed several times with different accounts.
Pretending to be different people. Using accounts to fake a dispute, a deal or the presence of separate players.
Escaping a punishment. A new account to get around a block applied to another account of yours.

If your account was suspended

When you open the game you see a screen telling you what happened, the reason, and how long the suspension lasts. Once that date passes the account comes back on its own: you do not need to ask for anything or talk to anyone.

That same screen has a field to ask for a review. Use it if you think there was a mistake, and there is a good chance there was: siblings, couples, flatmates and people playing from the same computer show up as linked on our side without doing anything wrong.

To get the review done fast, write in your request:

who you are;
which of the other accounts belong to other people;
how you are related (same home, same computer, same internet).

You can have one open request at a time. The team answers in that request, and the answer shows up on the same screen.

If you have two accounts and want to keep one

Move what matters to the account that stays yours through a trade or the market, with the usual fees. Then delete the other one.

To delete it, open Account in the game and scroll to the end, to Close the account. You type your trainer's name to confirm, no need to talk to anyone.

The account does not vanish right away. It stays locked for 7 days and only then is deleted. During those 7 days, when you open the game you see a Cancel the deletion button: one click and everything is back as it was. After that, it is forever: progress, pokémon, items, gold and Sapphire go away and do not come back.

The e-mail of a deleted account stays reserved for 7 more days after the deletion. If you want to start over with the same e-mail, you need to wait that long.

If you have an open Market listing, cancel it first: what is listed would be lost with the account.

There is no account merge. Level, pokémon, gold, Sapphire and Premium do not move from one account to the other automatically.

More than one trainer on the SAME account

This is allowed, and it is different from having two accounts: you play with one trainer at a time, and the others stay in the Dock, hunting on their own as if you had logged off.

It is up to four characters per account. Besides the main one, a slot comes from the account's Premium (30,000 gold, once), one from forming a Disciple in the Hall of Veterans (150,000 gold) and one from the Sapphire License (600 Sapphire, once). Each slot takes a new character or one bought on the Trainer Market. See how it works in Trainer License.

Sending an item from one trainer to another

In your bag, open the item and use Send to…. Pick the trainer and the amount: the item leaves yours now and shows up in theirs, instantly and with no fee. You can send it back the same way.

Three things worth knowing:

it only shows up when your account has more than one trainer;
a locked item (padlock) will not go - unlock it first;
a trainer listed in the Trainer Market cannot receive. Cancel the listing first, otherwise those items would go along to whoever buys the character.

Sending a pokémon from one character to another

In the Box, open the pokémon, go to the Market tab and use Send to another character. Pick the character: the pokémon leaves your Box now and shows up in theirs, with no fee.

What happens to it on the way:

its Bond goes back to zero on the character that receives it;
it follows the Badge of the character that receives it: a level 100 pokémon on a new character fights at that character's cap (trainer level + 20) until the character grows;
it does not count as received from another player: the Hall of Veterans, Ascension and Inheritance stay open for it.

What does not go:

pokémon on the team: put it in the Box first;
a locked pokémon (padlock): unlock it first;
a pokémon bound to its owner: the Founder companion, the Genesis companion and raid prizes stay with the character that earned them;
a pokémon on a tournament team or defending a guild gym;
a character listed on the Trainer Market cannot receive.

The full rule is in Game rules, in the clause about multiple accounts.