there are very complicated rules for whether you are tax resident or not in the UK - it depends on a number of factors to do with how many days you are abroad/in the UK plus a number of 'links' you might have to the uk e.g. owning a house, having a family here. These rules are nothing to do with being a non domicile.
A non domicile is 'largely' based on where your father considered his permanent home when you were born (though there are many other ways it works). In the true sense of it, it was designed for those who may have a genuine claim to 'belonging' abroad - i.e. may have a family home in another country and other earnings abroad but find themselves living in the UK. They still have to pay UK tax on what they earn in the UK - this is a fact that people seem not to understand - and a lot of these people are high earners in the UK (not all of them, but a lot of them) so do pay a lot of tax here. If they want to bring any of the earnings/gains they have abroad into the UK, they need to pay £30,000 to do so then they can bring what they like into the UK. The 'genuine' non doms I know are foreigners who have come over here to work but have large sums of money abroad - usually through working in other countries. In many cases they have paid tax abroad on these earnings anyway (in which case, there are other rules - some double tax agreements which mean they won't have to pay tax again anyway). The scrapping of the non dom rule won't really affect these people.
The ones it will affect are people who have money abroad they have never paid tax on, or lumps of cash not from work related income - so people with earnings in tax havens or large inheritances elsewhere. I imagine this applies to people like Lord Ashcroft (Tory) and a few of the large Labour donors too.
The thought before the election was that the Tories were going to raise the threshold - i.e. up the £30k they pay now to something like £100k. It came as something of a surprise that they didn't do this and Labour pounced and announced they were going to abandon it. It won't come as a surprise to any non dom and I suspect most of them have put some sort of tax planning in place in advance of the announcement anyway.
It may well affect the very very top end of the housing market and the market for luxury yachts ;) - but I don't think it will have an impact on any of us mere mortals.