If your business operates goods vehicles with a gross plated weight of more than 3.5 tonnes (or unladen weight more than 1,525 kg for unplated vehicles) then you will need a goods vehicle operator licence in England, Scotland and Wales. You will need a standard licence, which permits you to carry your own and other people's goods. You must have an operator licence for each Traffic Area in which you have a base. Licences are issued by the Traffic Commissioner for that area. There are eight Traffic Areas covering England, Wales and Scotland. In Northern Ireland you may need a goods vehicle operator's licence from the Transport Regulation Unit (TRU) of the Department of the Environment (DOENI). More information about goods vehicle operator licensing is available on the Gov.uk website - and from the DOENI in Northern Ireland.
It is an offence to drive, or to permit another person to drive, a vehicle on the road without a valid driving licence which covers that category of vehicle. To obtain a licence, drivers must take and pass a driving test for the appropriate class of vehicle. To drive a vehicle that exceeds 7.5 tonnes gross weight a driver must hold an LGV (large goods vehicle) licence. Drivers of dangerous goods vehicles are also required to hold a training certificate gained by attending an approved course and passing a written examination.
Drivers of large goods vehicles must hold a valid certificate of professional competence (CPC).
Note that anyone who transports live animals in the course of their business - for example a specialist reptile courier - requires a transporter authorisation from DEFRA and, where appropriate, a certificate of vehicle approval. You can apply for animal transporter authorisation on the Gov.uk website.
There are no other licensing requirements relating specifically to couriers, beyond the need for all vehicles to have a valid vehicle excise duty licence (road tax). Note that the law requires all of your motor vehicles to be insured all of the time unless they are declared to be off the road (SORN).