Because you are a British citizen, and because your mother is dependent on you, and because Ireland and the UK are both (still) in the EU, your mother can accompany you in Ireland under the European Union right of freedom of movement, as specified in directive 2004/38/EC.
To do this, she will apply for a visa, exempt from the fee, and after she arrives in Ireland, she will apply for a residence permit. Details can be found at http://www.inis.gov.ie/en/INIS/Pages/Family+Members+of+EU_EEA+_Swiss+citizens.
In particular, note the following:
If you wish to have your visa application considered under the Directive on the basis that you are a family member of an EU citizen exercising or planning to exercise free movement rights you must state this clearly on your application.
The directive provides, in article 6, for a "right of residence of up to three months" for union citizens and for "family members in possession of a valid passport who are not nationals of a Member State, accompanying or joining the Union citizen."
"Family members" are defined in article 2; the definition includes "the dependent direct relatives in the ascending line." This means that parents are included if they are dependent.
Article 7 concerns the "right of residence for more than three months," which applies (among others) to union citizens who "are workers or self-employed persons in the host Member State" and "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" also enjoys the right of residence for more than three months.
Most visa applications serve to authorize an applicant to apply for the privilege of entering the country issuing the visa, but in this case it concerns a legal right, which is much stronger than a privilege. Accordingly, the visa may only be refused under very limited circumstances. These are covered in article 5, which concerns the "right of entry"; it specifies that "such visas shall be issued free of charge as soon as possible and on the basis of an accelerated procedure."
Article 27 concerns the grounds under which a visa may be refused, which are limited: "Subject to the provisions of this Chapter, Member States may restrict the freedom of movement and residence of Union citizens and their family members, irrespective of nationality, on grounds of public policy, public security or public health. These grounds shall not be invoked to serve economic ends." This restriction of grounds of refusal means, for example, that you and your mother cannot be asked to meet any financial criteria as a condition of granting her a visa.