Fats are long chains of carbon atoms with hydrogen on each arm. C atoms have 4 arms so can either have another C on each side and H top and bottom, which is 'saturated' as all their arms are full. Or they can have only 1 hydrogen and hold onto a neighbouring C with a spare arm to make a double bond (unsaturated).
To generalise a lot, light oils are short chains with lots of unsaturated bits; solid waxy fats are long chains and totally saturated. Unsaturated is 'better' for you in as far as it's not associated with clogged arteries.
Saturated fats can be natural like fat on meat or in butter or cream, or unsaturated fats can be made more but not totally saturated by adding hydrogen to the spare carbon arms ('hydrogenated fats' like margarine). Some research shows that doing this creates different shape molecules than natural saturated fat (trans not cis, see above) and these are associated with more blocked arteries etc.
Thing is it's hard to tell if they're really that bad as trans fats are really cheap (coconut palm oil) and used in food that's not great in so many other ways (pastry, crumb coating, etc) and often bought by people who don't have much money so risk health problems anyway. Last I heard they were thinking of banning hydrogenated fat so presumably it is actually bad.
Note that unspecified 'vegetable oil' is almost always palm oil which is a lot more saturated than most veg oil, and often from palm plantations created by cutting down rainforest.
Conclusion: children need some fat, dairy fat not too bad in moderation, enjoy your icecream.