Let's break down this issue into its components:
1) cholesterol - butter is high in saturated fats, which traditionally was considered the leading cause of increased blood cholesterol. As we know, elevated blood cholesterol levels are a major risk factor for heart disease, in fact the leading cause last I checked. But the issue is not as black and white as that. In order to understand the idiosyncratic nature of saturated fats, one has to understand the various hydrocolloid chains in fats. I won't bore you to death with in-depth explanations about these, but suffice it to say that there is still no strong evidence either way in this respect. Studies have shown paradoxical outcomes for various lipid-based intake research. In fact they show vastly different results in men and women with various lipoprotein physiologies. All that we know, is that more research is required.
As an example of how complicated this research gets, men and women will use different indicators for HDL and LDL pointers for coronary disease or suppressed lipoprotein physiologies. Researchers for the last few decades did not realise this, and instead lumped everyone onto the same tests. We're slowly having to rethink and perform these tests all over again using better indicators. More info here if you're interested -
http://ajcn.nutrition.org/content/80/5/1102.full
Such conclusions make meta analysis of research more important, and such research has already been conducted -
http://ajcn.nutrition.org/content/early/2010/01/13/ajcn.2009.27725.abstract
The conclusion?
In other words, an analysis of all studies into this issue shows no correlation between higher saturated fat intake and increased risks of coronary disease. Again, going against what we thought we knew about heart disease. Here is the abstract from modern research using better indicators:
http://www.sciencedirect.com/science/article/pii/S0899900711003145
Other research shows no correlation between higher saturated fat intake and increased cholesterol at all, but rather a positive correlation between increased saturated fat intake and mass of LDL particles.
So the bottom line is that we have been conned by the margarine industry into believing outright that butter is bad for you. As far as we know at the moment, no specific case can be made either way. Some research shows a direct correlation with increased saturated fat intake and coronary disease, but modern research disagrees with a lot of the conclusions they came up with back then.
However that doesn't make margarine bad for you necessarily. In fact it can be pretty damn good for you:
2) Hydrogenation - let's start off with the basics. A saturated fat will usually be solid at room temperature. Like butter. Or lard. Or bacon fat. Unsaturated fats however will not be solid at room temperature. This is because of the way the hydrocolloids bond with each other and has to do with "spare" hydrogen atoms. Basically, unsaturated fats have double bonds that cannot stack neatly on top of each other, in simplistic terms, which means they cannot form a solid bond. Hence why they are liquid at room temperatures.
Hydrogenation is the process of chemically altering the structures of unsaturated fats to allow them to bond better, making them solid at room temperature. They achieve this using a combination of metal catalysts, heat and pressurised hydrogen injected into the oils. We have come to the understanding that hydrogenation is not good for you. Most modern research points to it being quite bad for you, in fact, and induces significant quantities of trans-fatty acids which are very bad for you. Hence why more than a decade ago already, most manufacturers turned their attentions to alternatives to hydrogenation for margarines and spreads. They came up with a process called interesterification. This is the process of moving fatty acids from one triglyceride to another, which in turn changes the viscosity of the product, its shelf-life, and its melting point, which allows it to remain solid at room temperature without the need for hydrogenation. So far interesterification appears to be perfectly safe, and it makes sense that it would be, as unlike hydrogentation, interesterification does not alter the fatty acids at all.
So nowadays most brands do not hydrgenate their oils to make margarine. In fact they use a perfectly safe method of altering the melting point of oils. In addition, the likes of Flora add cholesterol fighting additives to their spreads which have been proven to combat cholesterol. Flora in fact stopped hydrogenating their oils in the 90s already and I believe their current products are completely free of trans-fatty acids.
But be warned, cheap margarine manufacturers absolutely still use hydrogenated oils in their products. Buying the cheap stuff is just not healthy for you at all. Rather buy margarine that advertises that it doesn't use hydrogenated oils, and is trans-fatty acid free. There are a lot of butter proponents out there, and I in fact believe that butter is hardly a danger to your health unless you are eating hundreds of grams of the stuff daily. If all you do is put it on a sandwich then your intake is very low, and not a health concern. However there are some ridiculous websites stating that it is actually very healthy for you. Well, that's just not true. The truth is that we don't know yet - there is conflicting evidence in both directions. I'd say just keep it in moderation, or if you're concerned, swap to margarine which in this day and age (and from proper processors like Unilever's Flora) is actually pretty damn healthy for you...