In crypto-finance there are two main types of exchange. They serve the same basic purpose, but have a few important differences..
If you have ever used platforms like Binance or Coinbase, then you are familiar with the the traditional centralized exchange, or 'CEX'.
These are typically private, commercialized trading platforms, which operate above all to make a profit. They are usually regulated to some degree, which may vary based on region.
When you hold or deposit assets there, the 'wallet' is owned by the platform itself. This has advantages, and disadvantages. Advantages include multiple security features like 2-FA, high liquidity of a large selection of tokens and many trading options, from basic market buy/sell or limit orders to futures and margin trading. The main downside is that all of this relies heavily on trust. Trust in the platform and the people running it to remain honest, secure and solvent.
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More recently, a different type of exchange has been growing in popularity -- The decentralized exchange, or 'DEX'
The idea of a DEX as a viable alternative to CEXes is growing in popularity, as more people prefer full control and custody over their assets. The main consideration for users is this requires users to be more 'involved' with their trading, and take the time to research everything before getting involved. The inherent drawback to this is that your security is solely in your own hands - maintaining exclusive control over your wallet and seed phrase are of utmost importance here.
Most DEX platforms are trustless by design. Typically built with open-source protocols and running on decentralized infrastructure, they are resistant to unexpected control or manipulation by a single entity. That being said, they are not immune to exploits or development oversights.. but then again...neither are CEX platforms.