a miner has to connect to a node which is receiving transactions and verifying them.
then the miner selects from 0 up to as many transactions as fits in the block they are trying to create (usually near 2500 tx) based on their preference.
then the miner creates a hash from all these transactions they included in that block called merkle root hash.
finally the miner constructs a fixed 80 byte header from block version, merkle root hash, previous block header hash, nonce, block target, and block time. this input will be hashed using double SHA256 while changing nonce,... until the miner finds a hash that when converted to an integer in little endian format is smaller than the target.
when the miner succeeds the block is then broadcast to other nodes so that other miners also find out and stop working on the same block and start working on the next.
all those transactions that were included in that block in step 2 are considered "confirmed".
the block reward is also claimed by the miner in a new transaction without any inputs as the first transaction of that block called Coinbase tx.
this is the definition of how all cryptocurrencies using Proof of Work mining algorithm work. that includes bitcoin, litecoin, dogecoin, ethereum,....
How does it work out when a business wants to create its own digital coin? Does this mean they buy a data center and try to mine coins or what?
it depends on how they create that new coin. there is not just one method of creating a "crypto"currency.
