It depends on the residence permit you hold. I also assume you mean a short-term travel.
If it is the Article 10 residence permit (must have the wording "Residence card of a family member of a Union citizen"), you can accompany your EEA family member on their trip to Ireland without a visa.
If you hold a national residence permit, you'll need a type C visa:
Family members of EU/EEA/Swiss citizens seeking to reply on Directive 2004/38/EC (Free Movement Directive) – type of visa for which you should apply
If you are a non-EEA national:
- who does not hold a document called “Residence card of a family member of a Union citizen” as referred to in Articles 5(2) and 10(1) of Directive 2004/38/EC on the rights of citizens of the Union and their family members to move and reside freely within the territory of Member States, and
- wishes to accompany or join an EU/EEA/Swiss citizen family member who is moving to or residing in Ireland pursuant to the Directive 2004/38/EC,
You can apply for a single journey short-stay C visa which will permit you to enter and reside in the State for up to 3 months.
If you are a ‘qualifying family member’ of an EU/EEA /Swiss Citizen you are exempt from the visa fee.