A data structure describes how data is organized in memory.
Let's imagine a stack ADT with two operations: push and pop. We can decide to implement this ADT using, say, an array or a linked list. In either case, its operation will be the same (that is, you'll get the same results; but its time and space properties may be different).
I acknowledge that their names can be confusing (both contain the word "data"). "data structure" isn't a bad name; we understand that it's about how data is organized. But "abstract data type" is less clear, indeed.
edit: I think I found what may confuse you: I guess it's a visualization problem. When you're visualizing a data structure, it probably looks like boxes connected to each other with arcs. And I think your book has nice pictures for ADTs, too (stacks, queues, etc. drawn using boxes and maybe arcs as well). Whereas it's clear that an algorithm is very different from a data structure because it's described using pseudo-code. Forget the graphical representation for ADTs; think of them instead as a list of "rules" of the form "when I dequeue something from a queue, I get the last item that was enqueued".