Forgive my ignorance in this. I was provided some free space with which I can set up a blog, and I intend to get a domain to link it to the free space. Basically, I am curious about what happens after payment goes through for a domain of my choice. Do I get to link it to the blog address like the following example?
http://blahblah.com/blog1/index.html ==>
http://mydomain.com?
Yep, your new domain replaces the hostname of the current provider.
This assumes it's one of those free hosts which provides free space IF you register a domain through them. Most free hosts do not allow third party domain registration, you do it through their sales department and the domain might even be controlled by them or they'll try to sell you add-on services you don't need.
If so they usually have a control panel or web based area to set it up or they'll handle it all for you.
Please also be informed many operations like that make up revenue by 1) over charing domain registration fees 2) inserting ads in your site 3) poor customer service 4) bad web control panels
You might not care if it's a simple blog - just making sure you're well informed.
If you decide to switch:
If you end up buying a webhosting package which includes web space, domain registration and email usually there is an integrated control panel where you can set things up. The better ones set sensible defaults and all you need to do is configure email and upload your web files.
If you only register a domain and wish to maintain it independent of the webhost, you'll need to use the DNS zone file setup where you configure the zone with the IP adress of your web server, setup any other aliases you want such as ftp, email, etc.
I want to emphasize this is a VERY generic response covering two of the most common situations for newbies. If you need specific help with a webhost or domain registrar in particular, let us know, or use the help features on their own web sites.
Finally, a useful web page to help you understand the "larger view" of how this stuff works, written for those interesting in learning:
http://www.dotservant.com/knowledgebase/domain/
