The 'good fame and character' test focuses mainly upon the honesty of the applicant and their suitability to enter into a privileged relationship with clients.
Offences that are driven by dishonesty (for example fraud and theft) will count heavily against you, and there is a strong onus on the applicant to demonstrate that there has been sufficient rehabilitation and acknowledgement that the behaviour was wrong. Mitigating factors such as the persons age at the time of conviction and extrinsic circumstances are also taken into account.
Summary offences removed from dishonesty such as posession and common assault are usually not sufficient in themselves (dependent on the circumstances) to have a practitioner struck off, and so I would assume that similar offences would not kill all your chances of getting admitted in the first place. However, if the conviction occured within a short period of your application, it will likely count against you - particularly if you exhibited a total disregard for the authority of the law, or have not demonstrated any kind of rehabilitation.
Indictable offences such as murder and sexual assault are pretty much a guaranteed no.
Charges that did not lead to convictions (i.e. acquittals) are investigated by the admitting authority on their circumstances. There has been an acknowledgement - though I forget the exact case - that whilst some people have been found innocent, the circumstances of the case might reveal the applicant's unsuitability for practice. An example of this would be a man convicted of murder lying to the Police during the investigation in order to protect his wife - dishonesty, particularly when directed at the law, will kill any chance you previously had at being admitted. Of course, a substantial period of time/demonstrated rehabilitation can often defeat this.
In light of the above, it is vital that you lean on the side of 'excessive disclosure' when applying for admission (or so the textbooks say). If you are found to have not disclosed any prior convictions, charges, or mental health problems that effect you significantly, it will be seen as dishonesty and is prima facie enough for your application to be denied.