Assistant professor vs Associate professor
In a typical university in the United States:
An assistant professor is an entry-level faculty member. They are generally on the tenure track (although the term "assistant professor" does not guarantee this) but do not have tenure yet. Typically, within about seven years an assistant professor will either be promoted to associate professor or will leave the university, although the timing can vary a little and it's theoretically possible to remain an assistant professor forever.
An associate professor is one step up from an assistant professor. This promotion is usually the same as getting tenure, but not always. (Some universities, like MIT, frequently have non-tenured associate professors.) The final step for most faculty is a full professorship.
As for what an associate professor can do that an assistant professor can't, that varies even more than the terminology. In many US universities, the only additional power an associate professor has is voting on who gets tenure, but I wouldn't claim this is universally true.
In the Netherlands both assistant and associate professors are frequently tenured (= have a permanent position). Associate professors are expected to develop their own research line, while assistant professors can work on the topics of their bosses (full professors). Neither assistant nor associate professors can formally supervise PhD students: they can only co-supervise. There are some more minor differences: e.g., associate professors can be members of the Ph.D. assessment committee, assistant - not, unless they are co-supervisors of the candidate.
Update 2018: As of last year associate professors at some universities have been granted the right to formally supervise PhD students. Details of the implementation are however left to the universities, and, e.g., Eindhoven decided to grant this right only to senior associate professors.
In Australia, the typical hierarchy is:
- Level A. Associate lecturer
- Level B. Lecturer
- Level C. Senior lecturer
- Level D. Associate professor
- Level E. Professor
In the typical Australian ranking system, there is no "assistant professor" . In this academic ranking system associate professor is a high ranking. I think that both associate professor and professor in Australia would correspond roughly to professor in the United States.
As is noted in the comments, a small number of Australian universities have adopted (or adopted and then reverted back from) the American system. So you may find Assistant Professor is used occasionally. Assistant Professor probably maps onto Australian Level B (Lecturer).
Supervision of a PhD student depends on university regulations. At my university in Australia, there are several requirements in order to be a principal supervisor. In particular, (a) you need to have completed your own PhD or in rare cases be of equivalent standing, (b) have been an associate supervisor of PhD student to completion, or completed a set of training and experiential activities.