A blockchain wallet enables people to store and manage their crypto assets. Think of it the same as you would your bank account or your physical wallet.
There are two main types of blockchain wallets: hot and cold storage wallets.
An address can hold a balance of cryptocurrencies. A wallet controls a set of addresses. Think of an address like a lockbox, and a wallet as a key for many lockboxes. A wallet is accessible by providing a unique, secret 24-word passphrase. If you lose this passphrase, you don't have access to your wallet, and there is no way to recover your assets! Therefore, it's very important to safely store your wallet's secret passphrase securely. At the same time, anyone with your passphrase can access and take all of your assets, so it's vital to make sure nobody else knows your passphrase. It's best practice to not have your passphrase saved on any computer.
You can and should use a hardware Ledger to log into your wallet. Using a hardware wallet is the most secure way of accessing your tokens because your private keys and the passphrase never leave the device. If you have a number of tokens you would not be comfortable losing, it is strongly advised to invest in a hardware wallet.
Once you have your wallet, you may want to send your AVAX from an exchange to your wallet. See here for a guide on doing so.
Avalanche's Primary Network consists of three different chains, as explained in the overview article linked above. To move your funds from one chain to another, you will need to make cross-chain transfers.