There are many options when it comes to storing crypto, and all of them are either custodial wallets or non-custodial wallets. There are no exceptions, and you should know the difference so you can more efficaciously pick a means to your end.

Custodial & Non-Custodial Wallets

A custodial wallet is a wallet that has a 3rd party knowing the private key to a wallet that you are using, while oftentimes you do not know the private key.

A non-custodial wallet is a wallet that you know the private key to and often are the only one that knows it.

Now that it is clear what the simple difference is, let's get into the pros and cons.

Custodial Crypto Wallet

Pros

They are often easy for beginners. Don't have to remember the private key; just remember the account password.
Not remembering key/pass ≠ lost funds. Can recover it since 3rd party knows it. Risk of losing funds isn't on you.
Trading internally can mean no transaction (tx) fees since there's no actual tx occurring.
Tx can possibly be reversed if something malicious happens.

Cons

Don't have exclusive use of wallet. You are not your "own bank."
Account can be closed at any point for any reason.
Site can be hacked and funds lost, unlike cryptographic private keys.
Site can take funds are run with them (ex. Mt. Gox.)

Summary

While it is easier for beginners and people with lower tech skills, the control of the wallet is not exclusive to the person that has the account. Yes, the risk of losing funds via forgetting a code is greatly decreased, but the security of the wallet is jeopardized in other ways.

Many exchanges have this type of wallet for easier and faster trading. Two examples are Bittrex and Binance.

Non-Custodial Crypto Wallet

Pros

Exclusive control of wallet unless wallet "owner" shares private key.
Not waiting on 3rd party to approve tx.
Gives you an option of using cold wallets, which offer better security.
Hardware wallets are safe from even some malicious software since the signing of the tx happens offline.

Cons

Loss of private key = loss of funds. No way to recover other than finding or remembering key.
Can be expensive if hardware wallet.
Cold wallets are harder to trade with than hot wallets.

Summary

While the risk of losing the private key is on the person who made the wallet, one can be "their own bank" and have complete control of the wallet. The security of it is very high since the weak point is not a password found on a centralized website.

There are many types of this wallet including paper, hardware, and apps. Two well-known companies for hardware wallets are Trezor and Ledger, and a respectable app wallet is Ethos.

Which one?

This question can only be answered by you or somebody that has the relevant information for your situation and wants. What this article has aimed to do is to give you some cursory information and a few options.

The above references an opinion and is for information purposes only. It is not intended to be investment advice. Seek a duly licensed professional for investment advice.