It’ll be a DBS (Data and Barring Service/Disclosure Scotland check. Depending on the level (basic I’d imagine for your role), they’ll check for any criminal convictions. Even if you have a criminal record, it’s not an automatic bar to appointment, as long as you declare everything. Only recordable stuff will show up on the check you’re likely to undergo (by recordable I mean on the Police National Computer - so things like Motoring offences won’t, unless they’re criminal such as drink driving). I assume they sent you some paperwork to complete in order to carry this check out?
They will also check your references at this point I’d imagine.
They allow a couple of weeks for DBS etc, but they usually only take a few days to come back if all is well. References ca be another kettle of fish and it all depends on how proficient your referees are with this I’m afraid (some employers can take a while to get this sorted out, especially if the reference is from your current employer and they have to go through your HR dept.).