The suggestions above are great, however depending on the context (there might be a small difference) they can also mean the same. E.G. "Mi padre murió" - "Mi padre se murió" convey the same meaning.

Morir just means to die with no extra meaning or nuance. Mi perro murió de viejo. My dog died of old age. Morirse has many meanings. 1. To die suddenly (I believe this meaning can only be conveyed with the preterite tense) El hombre se deslizó del alicante y se murió. The man slipped off the cliff and died. 2. To have an intense sensation Me muero de hambre. I'm starving/dying of hunger (figuratively) 3. Extreme impulse/desire for something to succeed (from RAE) Se muere por ese cuadro. He is dying for that painting to succeed. (He really wants it to succeed) Se muere por ella. He is dying for her to succeed. 4. To pass away (euphemism) (I personally haven't seen it used this way much, but it's on WordReference) Se murió después de luchar años contra el cáncer. He passed away after struggling with cancer for many years.