So I know we have the wallets that our hosted by several companies both online and locally on a device.
Yes. A wallet is a computer program that maintains your list of private keys, scans the blockchain for transactions that supply you with value, and allow you to create and broadcast transactions onto the network.
We have the blockchain where transactions go through.
Go through? That's a rather misleading over-simplification.
Transactions are broadcast to connected peers on the network. Those peers verify that the transaction is valid, and then broadcast the transaction to their connected peers. Those peers verify that the transaction is valid, and then broadcast the transaction to their connected peers. And so on until almost every peer on the network has seen and verified the transaction.
Eventually some miners (or mining pools), which are also peers on the network receive the transaction and add it to the block they are working on. When one of those miners (or mining pools) complete the proof-of-work, they get to broadcast their completed block to all their connected peers to be added to the blockchain.
Are they just basically hashed numbers from one location to another.
No. They are a list of previously unspent transaction outputs that are used to provide value as inputs to the transaction, and newly created transaction outputs that encumber some value with a requirement that must be met if that value is to be used as an input into another transaction.
I mean what site is controlling the transactions and what if it were to go down permanently?
There is no site controlling anything. Therefore, there is no specific individual thing to "go down permanently". It is just a network of connected peers all agreeing to abide by the same rules.