Oliver Cromwell was not a Monarch but an English military and political leader and later Lord Protector of the Commonwealth of England, Scotland, and Ireland. As such his name did appear on coins. The actual monarch was Charles I, whom Cromwell was key in having executed.
Cromwell was an MP but after one year King Charles didn't recall Parliament for nearly 11 years. Eventually the chasm between King and Parliament led to a Civil War. Oliver Cromwell was a talented military leader and won several battles with the Royalists, eventually defeating them and putting the King on trial leading to Charles I being put to death.
Cromwell as Protectorate (1653-1658) was a controversial figure. He deposed the Monarchy yet replaced it with his own ruthless, tyrannical rule. He signed himself 'Oliver P' (P for Protector) as in the style of Monarchs; he was referred to as 'his majesty' and paid himself £100,000 per year, which was a lot of money in the 1650s.
He was offered the Crown for the Monarchy in 1657 to help get stability back to the country, but eventually turned it down. After a short illness in 1658 he died, and his son Richard became Lord Protector but that didn't last long and Richard resigned in 1659. The monarchy was restored in 1660 when Charles II returned from exile. Oliver Cromwell's body (or what they thought was his body) was exhumed and posthumously executed.