When talking about transformations, mathematicians often use the word "morphism". The word has a precise meaning in the branch of mathematics known as category theory, and a somewhat vaguer meaning elsewhere. In general, though, it's associated with the idea that when you study how one thing can be transformed into another thing, you should be particularly interested in transformations that preserve "structure".
A simple example is modular arithmetic. Take the addition table for the non-negative integers, and make from it another table where the addition "wraps round" whenever the answer is greater than a specific integer N. When N is 12, we have the familiar clockface arithmetic, where 11+1=0, and 11+2=1, and 5+7=0, and 5+8=1, and so on. The resulting addition table is wildly different from normal addition: for a start, it's finite. But it does have properties in common. For example, 0 is still special in that it does nothing when added. If we make a clockface-multiplication table, in which 2*6=0, and 3*4=0, and 3*5=3, and so on, then 0 is special there too, because multiplying by it still always gives 0. And so is 1 special: multiplying a number by it still gives that number. So making the clockface-arithmetic tables preserves the special rôles of 0 and 1. These are a vital part of the structure of the integers.
Outside mathematics, lots of examples occur in jokes. For example:
An American and a Russian were discussing politics. The American said "In our country, we have freedom of speech. You can stand in front of the White House and yell, 'Down with Reagan!', and you will never be punished." The Russian said "So what? I can stand in Red Square and yell, 'Down with Reagan!', and I will not be punished."In that joke, the White House and Red Square are like the 0 and the 1. They're a special part of the structure of the countries: distinguished elements which map onto one another. Reagan is another such element: the joke is that the corresponding element of Russia should be Gorbachev, so the transformation from America to Russia has violated the structure. In Metamagical Themas, Hofstadter has lots of examples (humorous and non-humorous) in the essays "Metafont, Metamathematics and Metaphysics", "Analogies and Roles in Human and Machine Thinking", and "Variations on a Theme as the Crux of Creativity". In the last, Hofstadter's "counterfactuals" are the transformations, and he's discussing whether they violate structure by "slipping" elements too far.
On rereading this, I notice that I seem to be implying that "structure" is about the distinguished parts that an entity has, and how those parts are preserved when it's transformed to another entity. But I don't think that's the entire story. For example, topologists study "structure-preserving" transformations between spaces, but many of those spaces don't have distinguished parts. Anyway, I hope the above gives an intuitive idea of what "structure" is. But as Michael Greinecker notes in his Stack Exchange answer to the question "Why do we look at morphisms?", it is hard to say what mathematicians do mean when they talk about structure. Neverthelsss, he has nicely answered the question I posed in my title, so that's where I'll stop today.