Basically, there are two kinds of wallet:
1. Hot Wallet
2. Cold Wallet

1. Hot Wallet:
                    A Hot Wallet is a wallet that is connected to the internet, and is used for day-to-day transactions. They are mostly used to store smaller amounts of Bitcoin as they are not too secure. Web wallets, Mobile Wallets, and Desktop Wallets are Hot Wallets as they are connected to internet and are used for usual transactions.

2. Cold Wallet:
                     A Cold Wallet is a wallet that has no internet access, and is used for storing Bitcoins for long-term. They are more secure as they need no internet access and are safe from phishing or any other kind of attacks. Hardware Wallets, Paper Wallets and sometimes Desktop Wallets (if not connected to internet) are used as Cold Wallets, to store Bitcoins for a longer time.

Now, each one of them has their own advantages and disadvantages. Like, Web Wallets has the disadvantage of Phishing or hacking attacks, and they often don't give you access to your Private Keys. Sometimes a wallet even can be a scam and run away with the Bitcoins of people. Advantage of Web Wallets is that they are user-friendly and fast too.
On the other side, Desktop wallets has no such disadvantages, but there are fears of keyloggers and other similar attacks. Their advantage is that they are safe from Internet attacks (Phishing, hacking), and the private keys are stored in your PC.
Paper and Hardware wallets can be lost, in such a situation, you will lose all your funds store in any of them, but they have the advantage that they are totally safe from attacks of any kind, meaning you are totally safe as long as you have them to yourself.

That is a small explanation of wallets. I hope I made them quite clear for you.