Quería comprarte un vestido, pero las tiendas estaban cerradas.
I wanted to buy you a dress, but the stores were closed.

El aula es pequeña, pero tiene mucha luz.
The classroom is small, but has lots of light.

Pero can also indicate that something is done despite the circumstances hindering the action:

No le gusta el trabajo, pero siempre es puntual.
He doesn't like his job, but he is always punctual.

¡Pero... may also be used to add emphasis.

¡Pero qué niño tan bueno!
What a good boy!

¡Pero si...! often indicates an objection or protest.

¡Pero si yo no he hecho nada!
But I didn't do anything!

Pero can also be used as a noun, in which case it's translated as "objection" or "defect".

Él siempre pone peros a nuestros planes.
He is always raising objections to our plans.

The conjunction sino is used to contrast the previous negative statement.

El cliente no quería mejores precios sino mejor sevicio.
The client did not want better prices, but (rather) a better service.

Use sino que when this conjuntion precedes a conjugated verb.

Pedro no enseña francés sino que enseña español.
As a noun, "sino" is "fate, destiny".