What is the name of the procedure of downloading and checking the ledger for a new node?
The state a full node is in while it is aware it is significantly behind the rest of the network is called initial block download, or IBD. Being in IBD makes the node make different tradeoffs, like preferring bandwidth over latency, and more aggressively batching operations.

While downloading blocks, getting a node closer to the most-work chain tip, block validation happens. This includes sanity checking the received data, verifying its difficulty and proof of work claims, checking that all coins spent by its transaction inputs exist and haven't been spent already, verifying the scripts and ECDSA signatures, and marking the inputs as spent and remembering the outputs created as being available for spending.

The whole process of learning and downloading and validating new blocks to get to the same knowledge other nodes have is called synchronization.