A blacklist is a list of IP addresses which have been detected as a source of spam. It has nothing to do with whether or not you can access the internet. However if you are running your own email server and you are blacklisted, you will find that some people will not be able to send email to you, or receive email from you. There are in fact many blacklists, each individually maintained - and it is possible to remove your IP from the blacklist.
A great tool I use often is MX Toolbox: http://www.mxtoolbox...blacklists.aspx
It will take your IP and scan a huge list of known blacklists and return your results in a single output. If you are blacklisted in any of them it provides a link you can follow where you can read more information about why your IP is blacklisted and what steps you need to take to remove it. The idea being here is that if you are not explicitly running a spam email server, then there is a problem with the configuration of the email server that needs to be resolved. Resolve the problem, then get the IP address removed from the blacklist. Most blacklists do not give much leniency. If you repeatedly ask to have your email server removed but don't make any changes eventually you will be blocked pretty much permanently. So if you are not an experienced email server admin, do not proceed lightly (in fact even experienced email admins would likely be extremely careful regardless).
There are ways to configure an email server to use a blacklist for security reasons. Thus if someone is trying to send you email from an IP address that has been blacklisted, your email server will reject the email along with a message back to the sender with a rejection code.
So in a nutshell:
1) Fix the email server(s) working on your IP addresses which are blacklisted
2) Remove the IP address from the blacklist in which it is listed