Going off of numbers alone, Yamato and her sister Musashi are very impressive ships. Their nine 46 cm (18 inch) guns were the largest guns to ever be fitted to any class of battleship. In comparison, the Iowa-class battleships, commissioned after Yamato and Musashi, stuck with 16 inch guns. Looking at armour, Yamato has thicker armour than the Iowa-class (the best battleships built by the US) just about everywhere - the belt (16.1" vs 12.2"), deck (9.1" to 6"), turret faces (25.6" to 19.7"), et cetera. It's no surprise that Yamato's displacement was so high - 71,659 tons fully loaded, compared to Iowa's 57,000 tons.

That is what makes them "super battleships."

However, numbers don't tell everything. Yamato is very impressive on paper, but many suspect that an Iowa-class battleship would get the better of her in an actual combat situation. Japanese steel was of mixed quality at best - no where near as good as that found in US ships - so whilst Yamato had thicker armour than Iowa, it wasn't necessarily as much better as it might seem at first. The 18 inch guns on Yamato could outrange the 16 inch guns of Iowa - around 45,000 yards vs 42,000 yards, but in reality, they would be unlikely to hit anything. American Radar-based fire control was much more accurate than anything that the Japanese were capable of, meaning that Iowa would be more likely to land hits on Yamato at range. Given that Iowa was also a few knots faster than Yamato, in a hypothetical engagement, she'd have the ability to break off of combat at will.

Yamato and Musashi were not terrible ships, though in hindsight one can easily say that they were a waste of resources (of course, this is all hindsight - in the 30s, naval leaders really did think that battleships would still play a vital role in fleet combat in the future). They weren't as refined or capable as the Iowa-class battleships, however.

Combinedfleet.com has a very good battleship comparison page, which runs down through various categories. It's worth a read.