You discover the information - by word of mouth, a shared forum, inspired use of a search engine, or plain luck...
You access (or copy, or read) the information - today typically fetching it over HTTP...
You use the information - think about it in your brain, reference it in your writing and speech, play it from your iPod, ...
...and at each point and for each step there are one or more - often Government-supported - mechanisms which may be deployed to try and stop you.
These mechanisms fall in and out of favour - to inhibit use there is DRM or license agreements backed up by legislation - but these don't work very well since they operate after the fact, only inhibit people who care about prosecution, and/or are technically hard to implement.
So governments are looking at the other two steps - regards discovery it's very plausible that the law leans directly or indirectly upon the likes of Google to prevent certain keywords being searched; in fact I'd be very interested to know whether or not these newspaper injunctions get served directly upon Google or whether the recent apparent gaps in Google's searchbox autosuggest results have accurately reflected the state of the web.
However regards Twitter the legal and moral precedents to inhibit discovery are complex, especially regarding the current, joyful #superinjunction hashtag; not only is Twitter hosted in the USA (ie: outside of UK jurisdiction) but it is also immensely popular, plus the entire tweetstream is available to the web and also to corporate subscribers - so any attempt at filtering superinjunction references from Twitter's own search engine is bound to be spotted and bypassed - thereby invoking the Streisand effect.
To quote a tweet by maniacpony: Hey kids, unsure if your favourite celebrity has a #superinjunction ? Just check if their #Wikipedia entry has been protected from edits...
Twitter is conversation, it is speech, and there doubtless will be a period of some discomfort whilst the UK's lawmakers grapple with this concept; I half expect someone to suggest that all UK Twitter users be filtered through a proxy to block, log and drop tweets containing particular words - the same tactic that China uses to repress dissent over Skype - which is also not too far removed from what the UK already does in the name of protecting children.
So finally we shall have a war on access - which is how the UK's IWF works (a blacklist of URLs) but is also what the US Department of Homeland Security's Immigration and Customs Enforcement (ICE) team is trying to similarly achieve through a series of domain name "seizures".
The government's thought process here is simple: if you can't look up the IP address of the machine which serves the data, then you can't get at the data - but they are wrong:
Homeland Security's ICE unit is not happy with a Firefox add-on that allows the public to circumvent the domains seizures carried out during the past several months. In an attempt to correct this 'vulnerability' in their anti-piracy strategy, ICE have asked Mozilla to pull the add-on from their site. Unfortunately for them Mozilla denied the request, arguing that this type of censorship may threaten the open Internet.
The software is startlingly simple: if the user attempts to navigate to a known "seized" domain name (eg: http://torrent-finder.com) then it looks-up a non-seized equivalent domain name and sends the user to that instead. This is hardly rocket science, but it is anathema to those who believe the DNS provides a potential point of control as opposed to being merely one of many address-lookup mechanisms.
How will government respond? No-one can say for certain. I've described other attacks and bypass scenarios, once in the context of IPv6 adoption and once regarding the unitary nature of DNS, and this Firefox plugin represents an interesting intermediate step towards the latter - towards the establishment of a "shadow" DNS. For further technical information - or just for the sake of understanding - go read the plugin's FAQ. Naming the plugin MafiaaFire - for Music And Film Industry Association of America (MAFIAA) and Fire (melts ICE) - might be thought a bit tactless, but since your correspondent once wrote some software called Crack he's really in no position to preach.
If you take nothing else away from this article, however, take at least this: governments are actively seeking ways to prevent you accessing any information that might disturb you.
It's probably not a good idea to let them do it.