As far as I understand, mining works like this.
Your understanding is missing some important pieces, and is incorrect about some of the pieces. You'll need to do some more learning. The posts in this thread do a good job of sending you down the correct path.
They get the "data" from the pool,
Are you talking about a mining pool? Or are you talking about a mempool?
If you are solo-mining, then you build your own blocks from the transactions that are in your own mempool (which your full node has built).
If you are mining in a mining pool, then the software run by the mining pool operator builds the blocks for you from the transactions that are in their mempool (which their full node has built). The mining pool software just sends you a mid-state (which is a partially hashed block HEADER), and you complete the proof-of-work on that mid-state.
in in order to verify them and put them in a "block",
If you are solo-mining, then your software builds the block and then sends the mid-state (the partially hashed block HEADER) to your ASIC equipment.
If you are mining in a mining pool, then the software run by the mining pool operator builds the block and then sends the mid-state to your software.
they try to find the nonce with which if hashed with the data and the previous hash data produces
the hash result that starts with 0000000000xxxxxx...xxx.
The block header consists of 6 pieces of information:
- Version
- Previous Block Hash
- Merkle Root (A hash that represents the transactions in the block)
- Timestamp
- Current difficulty target (frequently called "bits")
- Nonce
This data is partially hashed to create something called a mid-state. Then the "miner" tries to find a nonce which if used for the completion of the hashing produces a hash result that is LOWER THAN the value indicated by the current difficulty target.
What I don't understand is, what is this "pool" where they send the data to all the nodes?
That depends on if you are asking about the mempool or a mining pool.
Also, how do the nodes
verify the integrity?
Are you asking about how a miner verifies the integrity of the mid-state that it receives? Or are you asking about how all the nodes on the network verify the completed/solved blocks that they receive before they relay them to their peers? Or are you asking about how the nodes verify the integrity of the transactions before they relay them to the solo-miners and mining pools?
How the integrity is verified depends on what you are asking about.
What if the pool is compromised and sends the wrong data in the first place?
If it is a mining pool, then the participants will just try to find the nonce for the mid-state that they receive. The participants don't know or care if that mid-state is valid. All they care about is getting paid for completing the work. However, the mining pool will quickly run out of money if they are paying their participants and they aren't creating valid blocks. They eventually won't be able to afford to pay their participants, and the participants will all leave to find another mining pool.
If you are asking about the mempool of a solo-miner, then the solo-miner is responsible for running a valid node that verifies the transactions before it adds them to its own mempool. If it is compromised, then the solo-miner will create invalid blocks and he will never earn any money.
How is finding the right nonce alone guarantee the integrity of data and transactions ?
It doesn't.
Bitcoin is a trustless system. Every node on the network validates EVERY transaction and EVERY block that it receives. It NEVER trusts a block or a transaction that it receives from ANY other node regardless of whether the proof-of-work is complete or not. Each node will only add to its own blockchain the blocks that it independently determines for it self are valid.
Finding the right nonce provides a way for the decentralized nodes to all come to agreement on which VALID blocks to include in their chain, and which VALID block to reject. That way everyone is working from the same chain of blocks.
It also makes it extremely difficult and expensive to try to go back and modify ANY historical blocks.
