A (blockchain) wallet is a piece of software that enables access to the assets stored on a blockchain address.
In most blockchain systems, addresses are publicly viewable by everyone. In order to make a transaction from an address, you need to digitally sign the transaction with a secret ‘private key’ which proves that you are the owner of said address. An analogy often used is that of safety deposit boxes; each safety deposit box can be thought of as a blockchain address. In order to access the contents of the box, you need a key.
A blockchain wallet simply stores the private keys you need to access any addresses that the wallet controls.