I've just bought a host and it just has been set up. But hen it say this "In order to transfer your domain name onto our service
you must contact the current host and ask them to amend
the name servers for your domain as follows:" I don't understand what they want me to do. this is my first time.

I think that's for people who have another website elsewhere, and want to move their website to the host. You may need to double check exactly what you signed up for.

is that message from the "domain name" company or your "host space" company?

basically, they are two separate things. although the same company may supply both services to you at the same time depending on what package you sign up for.

if you have purchased these from two different companies, then they are just telling you to send the "domain name" company a bit of information to tell them where your space is located. your host space company should provide you with a set of name server numbers. you will enter these numbers in your domain name account so it knows where to point visitors when they type in your domain name.

perhaps do a quick search in this forum for domain name and host to get some other views that have been discussed before.