As had been written above, a criminal record is no bar to obtaining a passport. (35% of British men have a criminal record before their 30th birthday, so there would be a great many disappointed football fans, for example, if such a rule applied).
Anyone with a criminal record is barred from entering the USA with an 'ESTA' (Electronic System for Travel Authorization), whereby travellers simply provide relevant information online. They must instead seek a visa. That's a lengthy and complicated process, as it involves getting hold of a document from the police, together with court records, and attending an interview in London. The whole process can take several months (and there's no guarantee of getting a visa anyway).
Further, anyone convicted of an offence involving 'moral turpitude' (other than a single offence while still a minor) is automatically barred for life from entering the USA (although there's actually an appeal process - read on!). 'Moral turpitude', rather oddly in my opinion, includes all offences of theft but excludes most offences of violence. So you can be barred from entering the USA for nicking a single sweet from the pick-&-mix display but allowed in if you've beaten someone up!
If someone has been convicted of an offence involving 'moral turpitude' the US Embassy has no alternative other than to refuse them a visa in the first instance. However their case can then be referred to Washington as an application for a 'waiver of permanent ineligibility'. Such cases take ages to process and can still end in refusal. (There was a case reported here on AB where a guy had two convictions for driving without insurance. He had to wait 15 months to here whether he'd be given his 'waiver', only to be turned down).
Most other countries (e.g. all EU countries) don't require British passport holders to obtain a visa or, alternatively, their visa application processes doesn't ask about criminal convictions, so there can be no bar to travelling to such countries.