Monaghan is the county town of Co. Monaghan. That’s in the province of Ulster in the north of the island of Ireland. Monaghan today is in the Republic of Ireland, though obviously at the time your relatives were transported it was part of the UK. Tyrone is a county further north and is now in Northern Ireland (part of the UK).
Statutory records of birth, death & marriage records started in Ireland in 1864 (1845 for Protestant and Non-Conformist marriages). Prior to that you are heavily dependant on church records, where they exist. In general, there are few records prior to 1800.
Knowing only that your relatives came from Tyrone is not really sufficient to trace them that far back. Ideally, to trace an ancestor in Ireland through church records you need to know their religion, townland and parish. To find that information if the person died overseas, the usual sources are naturalisation records (where applicable), gravestones, family bibles, wills and obituaries.
For Northern Ireland, the Public Record Office (PRONI) has copies of many of the parish registers, and has a list of those held elsewhere.
www.proni.gov.uk/The records themselves are not on line and you need to search them in person. (PRONI is temporarily closed until April 2011 but there are temporary arrangements to view this data at Cregagh Library, Belfast. See PRONI website for details.)
For the Republic of Ireland the equivalent records are held by the National Library
www.nli.ie/If you have no other info about your relatives, your best bet might be to see whether their trial was reported in the local papers or if any court records have survived (though I suspect not).
Monaghan library may be able to advise you what papers were in circulation at the time, and they might even have copies. The National Library in Dublin is probably the place to start looking for court records (website above).
Elwyn