The term ‘wallet’ is used in Bitcoin, as well as other cryptocurrencies, for any software which the end-users uses to interact with the Bitcoin network, for example to make a payment or sign a message with their address.
The name is an analogy, making the comparison to a physical wallet that you may use to keep your bank notes and coins safe whilst carrying them around. For all practical purposes it will appear to a cryptocurrency user as if their wallet software actually contains their coins, as when they log in to the wallet they see their balance and can then spend their coins. This is not quite true. In fact, the coins are stored within the blockchain itself, which every part of the network has access to. You wallet is just used to interact with the network as to sign transactions with your private key, proving that you are the rightful owner of that particular address.
If you are looking for a wallet to use then please take a look at our Best Bitcoin Wallet List and Comparison Table