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 (12, 22 or 24 words).

Wallets, such as the Hedera Wallet (iOS / Mobile) or MyHbarWallet.com, are software tools used to access and control a unique Hedera mainnet account ID.  A wallet is similar to a "remote control", in relation to a Hedera account ID —  with the right credentials inputted into a wallet, they allow you to access and control a unique account ID on the network that holds HBAR cryptocurrency, even if just for a temporary amount of time (such as the case with 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.