'I'm having a party next week' means that you are throwing a party next week. Perhaps it's your birthday, and you've invited your friends to celebrate with you. You might say, by way of invitation 'I'm having a party next week and lots of my friend are coming. Would you like to come, too?' The present continuous form of the verb 'have' shows that it is a dynamic (rather than static) use of this verb. This makes it an action, indicating that it is you who are organising the party.
We use the present simple for fixed arrangements and plans, especially if you are referring to a schedule. So the only time you'd say 'I have a party next week' is if you are looking at your diary, and someone says 'Have you anything planned for next week?' and you might say 'I have a party next week' in the same way as you'd say 'I have an exam next week' or 'I have a dental appointment next week' - in other words, something in your schedule which you have to go to. Somebody like a caterer or a DJ might say 'I have a party next week', meaning that it is in their work schedule as something that they have to do.