Once they are on the system then yes, if they try to register again then it will complain at them. This is correct behaviour. When you clicked "remove" it didn't delete the account, but merely removed them as a member from your site.
If you invite them again from the members page then they should receive an email inviting them to join. It might end up in their spam folder if they have one, it would be worth checking just to be sure. If it definitely hasn't been sent then if you could let me know what the email address is then I will see if I can find out why (email me at admin@e-voice.org.uk rather than putting their email address on here).
Alternatively, if they log in and go to your site then they should see a "Request Membership" link a the top of the page. If they click that then you will be notified by email, and you should see them appear in the "needs approval" section of your members page.
Thanks
Joe - Voice Admin