Both dar and dejar can mean to give. The difference is that the verb dar means to give away, while the verb dejar means to lend:
¿Me das 15 euros? – Could you give me 15 euros (forever, without giving back)?
¿Me dejas 15 euros? – Could you give me 15 euros (for some time) / Could you lend me 15 euros?
Sometimes, these verbs are confused, especially when talking about "giving something to read, watch or listen":
¿Me dejas (ver) esta película? – Could you lend me this film (to watch)?
You cannot say ¿Me *das esta película? which would mean "Can you give me this film forever?"
It is easy to memorize it like this: dar with specific objects means "you’re giving your things away", regalar cosas mías:
A Juan le voy a dar mi jersey = A Juan le voy a regalar mi jersey
You can only use dar with abstract noun:
Te doy mi número de teléfono – I’ll give you my phone number