I am a citizen of Uzbekistan and I have a residence permit issued by Hungary. Do I need a visa to travel to Ireland?
Update: Ireland is in EU like Cyprus. Recently I have travelled to Cyprus without visa. This makes me confused.
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You can consult the web page of the Irish government, notably
Note: A Schengen visa or UK visa is not valid for travel to Ireland. For exceptions to this, please see Schedules 2 and 3 below.
Some exceptions exist for people who hold a UK visa (with very particular conditions!), but no additional exemption exists for Schengen or EU residence permit holders (except, of course, an Irish residence permit).
To answer the main question and for greater certainty, as of 2022, a Uzbekistan citizen requires an Irish short-stay visa, even if they hold a Schengen residence permit.
Ireland in EU as Cyprus. Recently I travelled to Cyprus without visa. This makes me confused.
As far as a traveller is concerned, EU membership does not matter unless you are an EU citizen. You need to check either your destination is within the Schengen Area (not all Schengen countries are EU and not all EU countries are Schengen), which applies a uniform border and visa policy and exempts any Schengen residence permit holder from short-stay visa requirements, or what the rule is for the individual country.
Each non-Schengen country decide itself (of course, with certain EU pressure) whether to allow Schengen residence permit to enter or not. Cyprus decided that it wants to exempt Schengen residence permit holders from needing a visa. Ireland did not decide so.
As part of the EU, Hungarian citizens (and other EU citizens) enjoy the freedom of movement and can without prior permission visit, work, study, reunite with non-EU family etc. in Ireland (or any other EU country). You are not an EU citizen and do not benefit from this.
Your Schengen residence permit exempts you from needing a short-stay visa for Schengen countries (including those not in EU, e.g. Switzerland and Iceland), not EU countries.
Several EU countries are not in the Schengen Area (as of 2022, Ireland, Bulgaria, Croatia, Cyprus, and Romania) and you must check if you need additional visa or not according to your nationality and other documents you hold for each of these countries. Vice-versa, most holders of a residence permit issued by e.g. Bulgaria or Ireland are not visa-exempt to visit the Schengen area.
As an aside (thanks to the comments), Bulgaria, Croatia, Cyprus, and Romania will join Schengen eventually and have started aligning their policies to the Schengen requirements. They may optionally recognize Schengen (and each other's) visas and residence permits as equivalent for short-stay purposes; until they join, you have do check the requirements individually.
Due to the special arrangement between UK and Ireland, Ireland has opt out of Schengen and will remain so for the foreseeable future barring major changes in the Irish situation or the UK foreign policy.