For instance, it was under the absolute rule of a person (Cromwell), and after his death the rule is passed to his son. When was it classified as a republic, as opposed to a monarchy? Was that before, during, or after Cromwell?
In short, because the means of power were dictated by democratic means.
During the time of Oliver Cromwell's rule England was a commonwealth that can be considered a republic because the people were represented in the government by elected officials. After the defeat of Charles I in 1653 the victors drafted the Instrument of Government (full text here), and afterwards Oliver Crowell was declared Lord Protector of the realm. However, the legality of the Instrument of Government was doubted, and in 1657 another group drafted the Humble Petition and Advice (full text here), which declared Cromwell Lord Protector for life, and gave him the ability to name his successor which is how his son, Richard Cromwell, became Lord Protector upon his father's death.
In sum, the "people" of England democratically set up the means by which Oliver Cromwell, and later his son Richard Cromwell, wielded their power. So, although the Cromwells exerted tremendous power they were propped up by England's two written constitutions.
Basically, a state is considered a republic if it has no king (or other type of monarch, i.e. duke, prince, count, atabeg, whatever). That's of course a narrow legalistic definition but that's the definition. As Drux has pointed out, there is no formal reason that the son of a ruler should succeed him, it just "happens", unlike in a monarchy where the line of succession is formally spelt out.
A Republic is not the same as a Democracy. It is quite unfortunate that the two main political parties in the United States are named Republicans and Democrats since between them, they have managed to corrupt the meaning of both words almost beyond recognition. A google search for 'define:democracy' gives:
with no mention of 'majority rule', while a search for 'define:republic' will give:
with no mention of 'Rule of law'.
The defining featutre of a Republic is that it is governed by rule of law (usually a constitution) and not by majority wishes. Whether or not the lawmakers are elected representitives depends on whether the country is also a democracy so places like GDR (East Germany as was), PRK (North Korea) and PRC (China) are all republics without necessarily being democratic.
A simple example of the difference between the two is that in democracy, a majority can vote banish a minority, but in a republic, that vote would be meaningless because of the protections granted by law.
In answer to the original question, the Commonwealth of England was a republic because it was governed by the laws of the land, regardless of the wishes of its leader.