I read this in many websites, but I'm still unable to understand the exact meaning/usage of...

That could mean different things depending on the context:

1.
to think or speculate curiously:
to wonder about the origin of the solar system.
2.
to be filled with admiration, amazement, or awe; marvel (often followed by at):
He wondered at her composure in such a crisis.
3.
to doubt:
I wonder if she'll really get here.
4.
it could also be used to ask something of someone in a polite way:
I was wondering if you could help me with this?