Controlling over 51% of the network and using their hash power to control the network and the transactions which get confirmed. By having over half of the total hash power, the miner can manipulate the network by orphaning confirmed blocks and selecting which transactions get confirmed or left in the mempool.
This in turns allows the attacking miner to selectively choose which transactions get confirmed - which can result in either no transactions being confirmed, specific transactions with betting outcomes, or executing double spends.
This type of attack is unlikely on any chain with significant hash power, but blockchains with low hash power are vulnerable to this attack.
Since the risk of a 51% attack decreases as support for a currency increases, the higher the hash power and the number of stakeholders for a chain, the more secure it becomes.
2.1 - BCHA
Shortly after the BCH / BCHA hard fork, the BCHA network was 51% attacked over the course of multiple days. During this time, confirmation times reached 6, 10, sometimes even 12 hours. This attack was mitigated when the BCHA node released an update to their mining client which stopped the attack.
* All terms and definitions may update as the Cryptionary improves.