How can I answer the question: "What do you hope to achieve during your studies at University X?" Currently I am writing (Statement of Purpose) for Phd applying. and they asked me this question:
 What you hope to achieve during your studies at the XX university? 
How I can answer this question, I need some ideas.
At this point, you can't really say what you will achieve or accomplish, exactly.  I mean, if you knew all that already, you kind of wouldn't need to go there to study, would you?

What I did was I described my fantasies, of what I wanted to study, which I hoped I would find there.  At the time, I was working as a COBOL programmer.  It was tedious.  Once in a while there was a fun logical challenge, but mostly I was pulling data from the student database at a community college, and writing tedious code to format the results in a nice report for a college administrator.

One fantasy was that I would write a program that would generate the tedious COBOL code.

That's not what I ended up doing once I was in the grad program.

But that was okay.

I don't remember what my other fantasy was, but I think it was something that I imagined as a result of programming a spline fit for a numerical methods class I took as an undergrad.  That was a really fun project and I wanted to do more of that kind of programming.

Just write about what sorts of things you'd like to learn more about when you are in a grad program.