Nothing will happen "under the hood" in the blockchain of a cryptocurrency when you trade on an exchange. The exchange has many wallets where the funds of the user are stored. The exchange has also a database where they store the information about the assets of the user, their trades and their resulting balance. Example:
A person A deposits 1 btc to exchange XY. Person B deposits 10 eth to exchange XY (these are transactions which you can see in the public blockchain of the respective blockchain).
Now the exchange has 1 btc and 10 eth and in their database they write analogous: "A has 1 btc; B has 10 eth".
Now person A wants to exchange his 1 btc into eth. So he buys 10 eth from person B and pays with 1 btc. This trade will not cause any transaction which will be added to the blockchain, because it is an exchange-specific process. When the trade is executed, exchange XY write in his database something like "A has 10 eth; B has 1 btc". (When person A, for example, now wants to withdraw his 10 eth to his ethereum-cold-wallet this would cause a transaction in the ethereum-blockchain.)
So when I initially transferred BTC from coinbase to binance to fund my trading account, does the coin go to my own personal binance wallet or binances's wallet? – ddrjca – 2018-01-18T21:25:33.630
1You don't have a "personal binance wallet". You do not have access to the private keys of the address that Binance gives, therefore you do not own the wallet. The deposit address that they give only to you is so they can register how much coins they owe you in their centralized database. They then display how much that they owe you in your Binance dashboard. – Cedric Martens – 2018-01-18T21:29:12.610