Llevar and traer are both verbs that indicate movement but it is important not to confuse them. 

Llevar is used to take something/someone somewhere from our location to another place. Traer is used to bring something from somewhere else to our location. 

Here are some examples:

Llevar

Llevé todas las cajas al otro almacén.I took all the boxes to the other warehouse.

Lleva este pastel a casa de la abuela.Take this cake to grandma's house.

Necesito llevar muchos libros al colegio.I need to take lots of books to school.

¿Me llevas a la estación por favor?Will you take me to the station please?

Hoy tienes que llevar a los niños al instituto.Today you have to take the children to school.

Notice that when it is used to take "people" to another location you need the preposition "a", like in the last example (a los niños). In addition, a second preposition "a" indicates "to" where (al colegio, a la estación).

Traer

Mamá, ¿me traes la toalla por favor? Mum, would you bring me my towel please?

Hemos traído unas botellas de vino para animar la fiesta.We brought some bottles of wine to liven up the party.

Javier ha traído a un amigo para cenar.Javier has brought a friend for dinner.

Laura nos trajo un regalo muy bonito de Brasil.Laura brought us a very nice present from Brazil.

¿Por qué traes comida de casa a la oficina? Tenemos una cantina con comida gratis.Why do you bring food from home to the office? We have a canteen with free food.

Note that if you are saying "to bring something somewhere"traer needs the preposition "a" to introduce where (a la oficina).

It is common to use the verb "traer" in bars and restaurants when we you ask the waiter for extra things. For example:

Camarero, ¿nos trae la cuenta por favor?Waiter, can we have [lit: can you bring us] the bill please?

Perdone, ¿me puede traer otra cuchara?Excuse me, can I have [lit: can you bring me] another spoon?

Traiga una jarra de agua para todos, por favor.Can we have [lit:bring] a jug of water for everyone, please?

It is very common to find indirect object pronouns with both traer and llevar to express to whom things are taken to or brought to (me traes, nos trajo, les llevó, te llevo...) as well as direct object pronouns expressing who one takes or brings something to, for example:

With indirect object pronouns

Les trajo un regalo.He/she brought them a present.

Me llevó las bolsas de la compra a casa.He/she took my shopping bags home for me.

With direct object pronouns

La trajo a casa en su coche.He brought her home in his car.

Nos llevará a un restaurante nuevo.He will take us to a new restaurant.

Bear in mind that llevar can also mean "to wear" or "to carry":

Lleva una falda roja y una camiseta blanca.
She is wearing a red skirt and a white t-shirt.

No puedes llevar tanto peso en tu maleta. 
You can't carry so much weight in your suitcase.

You can also watch this video:

 

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Examples and resources

Laura nos trajo un regalo muy bonito de Brasil.Laura brought us a very nice present from Brazil.
Javier ha traído a un amigo para cenar.Javier has brought a friend for dinner.
La trajo a casa en su coche.He brought her home in his car.
Mamá, ¿me traes la toalla por favor? Mum, would you bring me my towel please?
Traiga una jarra de agua para todos, por favor.Can we have [lit:bring] a jug of water for everyone, please?
Me llevó las bolsas de la compra a casa.He/she took my shopping bags home for me.
Hemos traído unas botellas de vino para animar la fiesta.We brought some bottles of wine to liven up the party.
Hoy tienes que llevar a los niños al instituto.Today you have to take the children to school.
Perdone, ¿me puede traer otra cuchara?Excuse me, can I have [lit: can you bring me] another spoon?
Les trajo un regalo.He/she brought them a present.
Lleva este pastel a casa de la abuela.Take this cake to grandma's house.
¿Me llevas a la estación por favor?Will you take me to the station please?
Camarero, ¿nos trae la cuenta por favor?Waiter, can we have [lit: can you bring us] the bill please?
¿Por qué traes comida de casa a la oficina? Tenemos una cantina con comida gratis.Why do you bring food from home to the office? We have a canteen with free food.
Llevé todas las cajas al otro almacén.I took all the boxes to the other warehouse.
Necesito llevar muchos libros al colegio.I need to take lots of books to school.
Nos llevará a un restaurante nuevo.He will take us to a new restaurant.
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