Whether you need a visa to enter Ireland depends on what country you're from.
There is a Short-stay Visa Waiver Programme in place for nationals of a number of Countries including Belarus, Russian Federation, Ukraine, Kazakhstan and Uzbekistan. If you have entered the UK on foot of a UK "C" General visa and have been granted leave to remain in the UK for 180 days, you may travel to Ireland within the time remaining on the UK visa, up to a maximum of 90 days (whichever is the shorter), without the need to apply for an Irish visa.
You may not need a visa to visit Ireland for less than 90 days if:
The categories of visas covered are short stay visit visas granted to persons seeking to leave to enter or remain in the UK except those seeking to enter as (i) a Visitor in Transit or (ii) a Visitor seeking to enter for the purpose of marriage or to enter a civil partnership.
Long-term visitors to the UK, for example long-term student visas, persons seeking to join spouse or family reunification, are not encompassed. If you do not qualify for the Visa Waiver Programme, you must have a valid Irish visa to travel to Ireland.
Belarus
Russian Federation
Serbia
Turkey
Ukraine
Bahrain
Kuwait2
Oman2
Qatar2
United Arab Emirates2
India
Kazakhstan
Peoples Republic of China1
Thailand
Uzbekistan
Yes you can. Once you are in the UK, there is no restriction on travelling to Ireland apart from normal airline security checks.
Yes, Jamaicans need a visa to enter Ireland.
yes i need visa to enter Ireland
Yes Americans do need a visa to enter the Czech Republic.
No.
They will need a visa to enter the United Kingdom in order to enter Northern Ireland.
yes there are no borders between northern and southern ireland
Yes.
Yes.
Yes.
Usually a separate visa is required to enter England than to enter Ireland, so depending on your own circumstances, if you need a visa to enter Ireland then you will also need another one to enter England. There are proposals from the Irish government to let people who have visas for Britain to enter Ireland without having to get another one for Ireland. As of the time of answering this question (June 2011) this has not been put into place. Even when it does, they doesn't mean that a visa to Ireland will be acceptable to England.
Yes.
Unless you are a citizen of a country that has a visa-free arrangement with Ireland, you cannot enter the country without one.