Book value per share takes book-value and spreads it across outstanding-shares. It rises with retained earnings and falls with losses, big dividend payments, or a share-buyback done above book value.
That last point catches people out. Buying back stock above book reduces book value per share even though it reduces the share count. Buying back below book raises it. Banks and insurers are judged on this closely.
Example: equity of $3.0B on 250M shares is $12.00 per share. The company buys back 10M shares at $20, spending $200M. Equity is $2.8B on 240M shares, or $11.67. Book per share fell by 33 cents.
Related: book-value, price-to-book, share-buyback, outstanding-shares