You are asking about EU freedom of movement, controlled by directive 2004/38/EC.
This directive defines several rights in connection with freedom of movement. These rights extend to citizens of EU countries and to certain of their family members. The rights extend to non-EU family members when the family member travels with, or to join, the EU citizen. From Article 3 (emphasis added):
- This Directive shall apply to all Union citizens who move to or reside in a Member State other than that of which they are a national, and to their family members as defined in point 2 of Article 2 who accompany or join them.
For the purpose of the directive, relatives in the ascending line qualify only if they are dependent on the relative from whom they derive their rights. (The ascending line is parents, grandparents, great-grandparents etc.) This answer assumes that your mother is in fact dependent on you, but you will need to demonstrate this to the Irish authorities in order for her to qualify.
Among the rights defined in the directive are
- a right of entry (Article 5)
- a right of residence for up to three months (Article 6)
- a right of residence for more than three months (Article 7)
The rights can be limited on grounds of public safety, public health, or public policy; this means that they can be refused only in rare circumstances. Furthermore, for visa nationals, the right of entry can be dependent on a requirement to hold a visa. From Article 5:
- Family members who are not nationals of a Member State shall only be required to have an entry visa in accordance with Regulation (EC) No 539/2001 or, where appropriate, with national law. [...]
However, this visa is supposed to be easy and quick to get, and it is free of charge. Paragraph 5(2) goes on to say
[...] Member States shall grant such persons every facility to obtain the necessary visas. Such visas shall be issued free of charge as soon as possible and on the basis of an accelerated procedure.
If the non-EU family member is entering after the initial three-month period, it may be necessary to show that the EU citizen is complying with Article 7 (see below). It is important to note, however, that if the EU citizen has not yet resided in the country for three months, it should not be necessary to do this. This follows from the simple definition in Article 6, quoted here in full:
- Union citizens shall have the right of residence on the territory of another Member State for a period of up to three months without any conditions or any formalities other than the requirement to hold a valid identity card or passport.
- The provisions of paragraph 1 shall also apply to family members in possession of a valid passport who are not nationals of a Member State, accompanying or joining the Union citizen.
If you plan to bring your mother after three months, however, you may need to show that you are in compliance with Article 7, concerning the right of residence for more than three months:
- All Union citizens shall have the right of residence on the territory of another Member State for a period of longer than three months if they:
(a) are workers or self-employed persons in the host Member State; or
(b) have sufficient resources for themselves and their family members not to become a burden on the social assistance system of the host Member State during their period of residence and have comprehensive sickness insurance cover in the host Member State; or
(c) [concerns students; omitted because it does not apply to parents]
(d) are family members accompanying or joining a Union citizen who satisfies the conditions referred to in points (a), (b) or (c).
- The right of residence provided for in paragraph 1 shall extend to family members who are not nationals of a Member State, accompanying or joining the Union citizen in the host Member State, provided that such Union citizen satisfies the conditions referred to in paragraph 1(a), (b) or (c).
The most likely paragraph to apply to you is (1)(a). Namely, you are likely to have a job or be self employed. If you are wealthy or retired and can afford not to work, paragraph (1)(b) applies instead.
You have expressed some interest in returning to the UK using the Surinder Singh route; a full discussion would warrant a new question, but this fact sheds some light on the question
do I need to apply for residence card or open a bank account which determines that I have now settled in an EU country and can exercise my EU Treaty Rights?
Essentially, you're exercising your rights just by entering the country. Your mother can join you in Ireland as soon as she gets a visa. You can even get her a visa before you move to Ireland, and she can arrive with you or anytime thereafter.
However, for you to move back to the UK with your mother, the British authorities will want to see that you were working (or otherwise covered under Article 7), so keep all of your pay stubs.
