cis-trans notation refers to the orientation of functional groups within a molecule. The answer here refers specifically to fatty acids (as a constituent part of a fat) but cis-trans notation applies to far more molecular forms than just fats.
In a saturated fat all of the bonds are single and so the chain is free to rotate. As a result the hydrogen atoms attached to the carbon generally move to adopt the lowest possible energy position, with adjacent pairs of hydrogens on opposite sides of the chain.
In mono and poly unsaturated fats the presence of double bonds stops any rotation at the double bond site. And so there are two possible orientations for any given double bond:
Adjacent hydrogen bonds on the same side of the chain; a cis fat, or
Adjacent hydrogen bonds on opposite sides of the chain; a trans fat.
Notice that the only difference lies in the 180° rotation of part the chain.
If there is more than one double bond then there can be any combination of cis-cis, trans-cis, cis-trans or trans trans. A trans arrangement tends to straighten the chain out so that it looks more like a saturated fatty acid whereas a cis arrangement introduces kinks into the chain.
In nature most, though not quite all, mono- and poly-unsaturated fats are of the cis form. The change from a cis to a trans arrangement can however happen with catalysed hydrogenation in the following way.
Firstly hydrogen gas (H2) is adsorbed on to the surface of the catalyst. The molecular hydrogen rapidly splits in to adsorbed atomic hydrogen which is free to migrate across the surface. 
Secondly the cis fatty acid is adsorbed on to the surface of the catalyst. This breaks the double bond but the rotation of the chain is not possible as both sides of the bond are locked to the catalyst surface.
Next one of the carbon atoms binds to an adsorbed hydrogen atom. The chain is now free to rotate and is likely to do so as this allows the molecule to adopt a lower energy configuration. So far all the steps are the same as for the normal catalysed hydrogenation reaction.
The next step is different as it reverses the previous step, however it does not reverse the rotation.
In the last step the second carbon is released from the catalyst and so the entire, newly modified trans-fatty acid is released back in to the reactor.
If the catalysed reaction is allowed to continue for long enough all the trans-fatty acids will be destroyed since the final product will be a uniform, fully saturated fat. However it is known that diets high in saturated fats lead to an increased risk of heart disease and stroke and so much of the oils treated in this way are only partially hydrogenated (you may for example hear of spreads that are "low in saturates and high in poly-unsaturates"). Paradoxically trans-fatty acids (generally described as ‘trans fats’) have been linked with lowering high density lipoproteins (HDL or ‘good cholesterol’) and raising low density lipoproteins (LDL or ‘bad cholesterol’). This in itself raises the long term risk of coronary heart disease.
