Normally in a criminal trial the jury or bench of magistrates cannot be told of the defendant's previous record before they consider their verdict. However, with the introduction of the "Bad Character" (BC) provisions they may be told (under strict criteria and only under one of the seven "gateways" I mentioned in my earlier answer). This essentially is evidence that which is not related to the offence or incident itself but to an individual involved in the proceedings - usually the defendant but sometimes a witness. In the case of the defendant the prosecution might want to introduce BC evidence in the form of the defendant's previous record. For example, "He is an habitual shoplifter; he has 99 precious convictions for theft from a shop; he has a propensity to commit this type of crime and his record demonstrates that. You should take this into account when deciding your verdict". They may also want to introduce it on the basis that he has a tendency to tell lies and so cast doubt on his version of events. You can read all the circumstances and the tests for BC Evidence to be admitted in the link that I provided but my brief summary above probably explains it well enough.
The admission of BC evidence is a matter for the judge (in the Crown Court) or the Magistrates (in the Magistrates Court). In the lower court the BC application is usually heard before a different bench of Magistrates to avoid the problem of the trial bench hearing of BC, refusing to admit it and then having to "put it from their minds" when deliberating. There are no hard and fast rules but as I said earlier, a single conviction is unlikely to lead to BC evidence being allowed. Any application has to be backed up by facts - the prosecuting lawyer simply disliking the look of somebody is not sufficient.
How this ties in with this question is this: in these cases of multiple offences the jury is effectively hearing evidence of the defendant's alleged bad character by stealth. They hear the first witness's allegations (on which they must later decide) then the second, who describes a similar but unrelated incident, then the third and so on. By the time they reach number ten their minds must be poisoned with the evidence of the earlier nine witnesses. They have heard of the defendant's - albeit unsubstantiated at that stage - bad character. The defendant has not been given the benefit of the protections under the bad character provisions.
"Hearsay" evidence is something completely different. It is "second hand" evidence. Witness A sees defendant D hit victim V. Witness A can provide first hand evidence of the incident. Witness A's friend, F, did not see the incident but witness A tells friend F that he saw D hit V. If called to give evidence Friend F can only say that A told him that D hit V. That is hearsay evidence ("somebody told me...") and is not usually admissible in criminal proceedings.