Your Hedera mainnet account ID (0.0.XXXX) is a unique account stored and accessible on the Hedera public ledger. With the right credentials, as explained below, you can control a specific account ID (send hbars from the account ID, make and pay for Hedera network service API calls, etc.).
The credentials needed to control a unique account ID are its public / private key pair or a mnemonic recovery phrase (22 or 24 words) — these can all be found in your Hedera Wallet if you're using it with an active Hedera mainnet account ID.
Wallets, such as the Hedera Wallet (iOS / Mobile) or MyHbarWallet.com, are software tools which can be used to access and control a unique Hedera mainnet account ID. A wallet is sort of like a remote control in relation to an account ID — with the right credentials inputted into a wallet, many of them allow you to access and control a unique account ID on the network, even if just for a temporary amount of time (such as in the case of MyHbarWallet.com).
You must securely document your public / private key pair, mnemonic phrase, and account ID — it is the only way in which you can control your account ID through various wallets or by using the Hedera API.
Note: Not all wallets will let you control an existing account ID using your credentials, but MyHbarWallet.com does today, and we anticipate more in the future.