A wizard can use a borrowed spellbook to prepare a spell she already knows and has recorded in her own spellbook, but preparation success is not assured. The reader must make an Arcana check (DC 15 + spell's level) to prepare the spell. If the check succeeds, the wizard can prepare the spell. She must repeat the check to prepare the spell again, no matter how many times she has prepared it before. If the check fails, she cannot try to prepare the spell from the same source again until the next day.
Wizards can add new spells to their spellbooks through several methods.
Spells Gained at a New Level: Wizards perform a certain amount of spell research between adventures. Each time a character attains a new wizard level, she gains two spells of her choice to add to her spellbook. The two free spells must be of spell levels she can cast. If she has chosen to specialize in a school of magic, the wizard also gets a third spell from that school.
Spells Copied from Another's Spellbook or a Scroll: A wizard can also add a spell to her book whenever she encounters one on a magic scroll or in another wizard's spellbook. She must spend one hour per spell level studying the spell. At the end of her study, she must make an Arcana check (DC 15 + spell's level). A wizard who has specialized in a school of spells gains advantage on the check if the new spell is from her specialty school. If the check succeeds, the wizard understands the spell and can copy it into her spellbook. The process leaves a spellbook that was copied from unharmed, but a spell successfully copied from a magic scroll disappears from the parchment.
If the check fails, the wizard cannot understand or copy the spell. She cannot attempt to learn or copy that spell again until one more point is added to Arcana. A spell that was being copied from a scroll does not vanish from the scroll if the copy fails.
Independent Research: If a library, arcane school, magical archive, or other source of written knowledge is available, a wizard also can research a spell there. To do so, you spend 1 day per spell level dedicated to study, after which you may your arcana check (DC 15 + spell level). If successful, you learn the spell you are researching (or one chosen by the GM if you don't have a particular spell in mind). If you fail your check, the GM may say you come up empty, though she may also say you learn a lower-level spell instead (assuming you check is good enough).
A wizard can use the procedure for learning a spell to reconstruct a lost spellbook. If she already has a particular spell prepared, she can write it directly into a new book. Unlike casting, the process wipes the prepared spell from her mind. If she does not have the spell prepared, she can prepare it from a borrowed spellbook and then write it into a new book.