verleihen
to lend
To make something available to someone for a certain period of time, often for a fee.
Die Stadtbibliothek verleiht auch E-Books.
The city library also lends e-books.

to lend / award / give
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to lend
To make something available to someone for a certain period of time, often for a fee.
Die Stadtbibliothek verleiht auch E-Books.
The city library also lends e-books.

'Verleihen' is the inseparable form of the verb 'leihen' (to lend/borrow). The prefix 'ver-' here emphasizes the action of giving something away or leaving it to someone else.
Kannst du mir bitte deinen Stift leihen?
Can you please lend me your pen?
The verb 'verleihen' almost always requires two objects: the person who receives something is in the dative, and the thing being lent, awarded, or given is in the accusative. The structure is: someone 'verleiht' [person in the dative] [thing in the accusative].
Die Universität verleiht dem Professor die Ehrendoktorwürde.
The university awards the professor an honorary doctorate.
The accusative object reveals the meaning: if it is an object (book, car), it means 'leihen' (to lend). If it is an honor (prize, title), it means 'zuerkennen' (to award). If it is a quality (shine, touch), it means 'geben' (to give).
Ich verleihe meine Werkzeuge nicht gerne.
I don't like lending my tools.
In the meaning 'zuerkennen' (to award), 'verleihen' is always combined with official honors. Typical phrases are: 'einen Preis verleihen' (to award a prize), 'einen Orden verleihen' (to award a medal), 'eine Auszeichnung verleihen' (to confer an award), or 'einen Titel verleihen' (to confer a title).
Der Nobelpreis wird jedes Jahr im Dezember verliehen.
The Nobel Prize is awarded every year in December.
In this meaning, the subject is often a thing or an abstract concept that gives a place or situation a particular quality. The structure is: [thing] 'verleiht' (gives) [something/someone] [a quality].
Die vielen Kerzen verliehen dem Raum eine festliche Atmosphäre.
The many candles gave the room a festive atmosphere.
to give
'geben' and 'verleihen' can both mean that someone receives something. But 'geben' is much more general: you can give an object, information, or an opportunity. 'verleihen' is more specific. It fits when someone receives something only for a certain period of time, when an award is officially given, or when something gives a thing a special effect. In everyday life, people usually say 'geben' when it is simply about handing something over. 'verleihen' sounds more natural when lending, prizes, or special qualities are meant.

Die Stadt verleiht dem Filmteam einen Preis für seine Arbeit.
The city awards the film team a prize for their work.
Der Arzt gibt dem Patienten heute einen wichtigen Rat.
The doctor gives the patient important advice today.