What happens if a miner deliberately ignores a transaction, and "wins"?
Miners have complete perogative over what transactions to include. They may include random transactions, the highest fee paying transactions, just their own transactions, or no transactions at all.
If a miner doesn't include a particular transaction from their mempool, it stays there. It will be available for later miners to pick it up and include in their blocks later.
To incentivize miners to include your transaction, you can include a fee. Miners in practice will pick the transactions with the highest fee first. First matters, because blocks have limited capacity, and so there may be more transactions on the network than can fit in a block.