Bitcoins only exist on the Bitcoin ledger. They never leave. All that changes is who has the cryptographic keys to control them.
People typically acquire satoshis from third-party exchanges. However, this step alone does not eliminate counterparty risk, because any relevant Bitcoin keys are held by that third party – not you.
A second step – you taking custody of your own satoshis using your own keys – is needed to make them truly yours. This may sound daunting, but is usually straightforward.
To perform this second step, you need a ‘wallet’. ‘Wallet’ options include dedicated hardware (for offline ‘cold storage’), software (which sits on a mobile device or computer), ‘multi-sig’ (where more than one person/ device is needed to gain access to the bitcoins), and even hardware/ software combination solutions. None of these 'wallets' actually contain your bitcoins (remember they always stay on the Bitcoin ledger), but instead control the private keys which grant access to them.
Which ‘wallet’ you choose will depend on your own circumstances, attitude, budget and knowledge level, as well as the value involved. But it should always be based on open-source software and have easy-to-implement and robust backup procedures, so you can still access your bitcoin(s) if your hardware/ software – or the company which made it – fails.
Until you take this second step, ‘your’ bitcoin isn’t yours at all: it’s just a balance owed to you by your third-party provider, which you may well lose (if the provider becomes bankrupt, for example).
Bitcoin is not truly yours until you make it yours.