Transferring a domain name depends upon the domain extension.
If the extension of domains is .uk, then we new registrar will provide his IPS tag. You will have to pass on the same IPS tag to current domain registrar and ask him to change it to the new one. Once he changes the IPS tag, your .uk domain name will be transferred.
If the extension of domains is .com, .info, .biz (other than uk domains and .eu extension), then you will need to ask your current registrar for EPP key for the domain name and pass on the same to the new registrar to initiate the transfer process. Once you provide the EPP key to the new registrar, he will initiate the transfer process.
If the extension of domains is .eu, then you just need to contact the registrar where you wish to transfer the domain. He will initiate the domain transfer on your behalf.