When peers disagree, how does the network decide which version of the ledger is correct?

Majority rules: every node compares its latest block hash against its peers, and any node out of sync with the majority is ignored — the version most of the honest network agrees on wins.

Because the tip block's hash encodes the entire history behind it, two nodes can compare a single hash to check whether their whole chains match. Suppose one peer maliciously edits a past block and even re-mines to make its own chain internally valid — its latest hash will still differ from everyone else's. When nodes compare, the majority holding the matching hash simply reject the outlier: it has effectively forked off onto its own private chain and is excluded from the shared ledger (and from any mining rewards). This democratic "majority of the network agrees" rule is what removes the need for any central referee. It is also why decentralization is a security property: the more independent nodes there are, the harder it is for any faction to be the majority.

Go deeper:

• Fork (blockchain) — Wikipedia — what happens when peers hold divergent chains and how the network reconciles them.