A wallet is a software program that stores public and private keys, and allows you to interact with blockchains to sign transactions that may include sending and receiving currency, or executing smart contracts. They don't actually store any currency; they can only reference transactions on the blockchain tied to that address.
Let's talk about public and private keys for a minute as it's an important concept to understand, because if you compromise your private key you will have compromised everything associated with it. Each wallet has a private and public key: think of the public key as a street address to your house, and the private key as the key that can unlock the mailbox. Anyone can send something to your mailbox using your public key, but only someone with the private key can unlock the mailbox:
Let's throw out a few more analogies of this, and then summarize a key point. Your email address is public and the password to your email account is private:
Your bank info is relatively public, but you have to provide some form of identification to use it:
Your car is publicly viewable, but you have to have the key to unlock it; maybe that particular car is a bad example, you know that might not even be the right key for it, but my point's still valid:
In all of those scenarios, if you lose your private key you can regain access to your asset by either calling a locksmith, talking to tech support, or going to the bank to get a new debit card. In blockchain technologies though, if you lose that private key, it's over; there's simply no way to recover a private key that's been lost, and that's by design. Any scenario where you can regain access once lost is because there's a middleman somewhere who still has access, whether that's the bank, the locksmith, or the email provider, and you pay for that convenience. With blockchain, there's no middleman and you own responsibility of your own account; the benefit of this ownership is your account can't be hacked unless you allow it to happen, and you don't pay a third party transaction fee for the convenience of using your money.