A wallet is used to store the keys to your cryptocurrency, regardless of where you purchase the crypyo (be it Kraken or any other exchange).
When working with an exchange such as Kraken, you already have you crypto stored in wallets. Unfortunately, you do not have the private keys to them and so, therefore, are vulnerable to hacks such as what happened to MtGox some years ago.
In addition, there are some things that you cannot do such as participate in most ICOs since they require that you hold the private keys to your addresses.
If you want to have full control, you need to have both the private keys and public keys to your wallet. I assume this is the case, so I would recommend purchasing a hardware wallet such as Ledger Nano S or similar (only purchase from legit original store). Hardware wallets are much safer than software downloadable wallets.
What you have to know beforehand though is whether the Wallet you decide to use supports your crypto or not. Legder product page details for example what crypto it supports. On top of that, all ERC-20 tokens are also supported.
If this is all fairly new to you, I suggest you search through posts to find the conceptual background you need first, before moving on to using the wallet you decide to use.
Start by reading this post:
https://bitcointalk.org/index.php?topic=2852255.0