Transactions and applications will still continue as normal on Ethereum, maintained by the same software clients we use today. In this sense, the terms Eth1 and Eth2 actually represent different layers of the stack: execution and consensus.
Execution clients (Eth1 clients). Responsible for transaction bundling, execution, and state management. This layer is already represented by clients that Ethereum uses daily, such as Geth, OpenEthereum, Hyperledger Besu, and Nethermind.
Consensus clients (Eth2 clients). Responsible for the validating blocks, also known as the Beacon Chain. This consensus-layer has been running smoothly for more than 6 months, with 155,663 validators collectively staking 4,981,046 ETH. Client teams include Teku, Lighthouse, Nimbus, and Prysm.