A "proxy avoidance and anonymizer" would be a server which sits between you and where you and the site you are trying to reach (but have been barred from) and gets the information for you, bypassing the block. It would also filter out some information about you to the server making you more difficult to track.
- These sites are acting as anonymous proxies.
- Privoxy springs to mind. Squid can be configured to do this, and pretty much any other type of proxying. There are no doubt plenty of other services - and indeed it would not be that hard to write your own variant using curl and PHP.
- You need to know that if you are doing something worth scrutinising you will most likely get caught - moreso if you need to ask the question, as there are a huge number of traps and gotchas.
There are alternatives which are probably more secure and less work which you should investigate, including using a VPN service, ToR, Tails - All of these are useful, but all have their drawbacks.