A simple cryptocurrency wallet contains pairs of public and private cryptographic keys. The keys can be used to track ownership, receive or spend cryptocurrencies. Your tokens are stored on the blockchain, but you can access them by using your private key.
A wallet, in this case, is just a user-friendly interface that helps you interact directly with the blockchain: get access to the tokens, encrypt information, sign transactions.
By exporting a wallet, we mean exporting your private key from your old wallet and using it to get access to your funds via the new wallet.
Some wallets provide not a key but a Seed Phrase (Mnemonic Phrase).
Seed Phrase is a list of keywords that the wallet transforms into your private key.
Some wallets for better protection require a seed phrase combined with other information, for example, a password or email.
They use this info to generate a more secure private key. So if you want to export your account from such a wallet, you will need to provide your seed phrase and other information.
For example, the Temple wallet can restore access to your funds via a Seed Phrase or a Private key. If your old wallet required a password you would have to provide this info too. Otherwise, the Temple wallet will generate a new private key that provides access to a new account. You will see the empty balance of the new wallet. To get access to your old wallet with funds, you will need to provide the password or other info. As an example, check how to restore the Tezos fundraising account.