Tenure and faculty rank are somewhat independent in that a faculty member may be hired as an associate (or even full) professor without immediate tenure- in those cases the faculty member is typically awarded tenure after a shorter than usual probationary period of a year or two.
The more common situation is when a faculty member is hired as an untenured assistant professor and then awarded tenure and simultaneously promoted to associate professor after a fairly long (up to seven years) probationary period. I'll focus on this common case in the US.
Does your salary increase automatically?
There is typically a pay raise given to faculty who are promoted, but this could be small (a couple of percents) or large (twenty percent) depending on the institution. There might be a standard pay raise, or it might be determined on an individual basis. At institutions with unionized faculty that pay raise would be part of the contract.
Does your teaching load change?
At some institutions, tenure-track assistant professors have a reduced teaching load which goes up when they're promoted. It would be very unusual for the teaching load to be reduced due to promotion and tenure.
Do you get (additional) grad students?
Graduate students in the US are typically supported by research assistantships (which are under the control of the faculty members who are PI's on the grants that fund the assistantships) and teaching assistantships (which are awarded to students by the department.) Students typically ask to work with an academic advisor and then either get an RA from the advisor or are supported by a departmental teaching assistantship if they can get one.
Thus promotion to associate professor doesn't really have any direct effect on your ability to support graduate students.
However, students may be more interested in working with you after you've been tenured. Students are rightly concerned about starting to work with an inexperienced assistant professor who might not get tenure and leave them with a half-finished thesis. Another issue is that in some disciplines (pure math is an example) it's traditional for assistant professors to focus more on their own research than supervising graduate students. In that situation, a newly tenured professor might be more willing to take on students.
Are these changes negotiable?
It's always possible for a faculty member to ask for a pay raise or a reduction in teaching load. However, you're unlikely to be able to negotiate a pay raise or different working conditions just because you've been recently tenured. The norm is that faculty members who are tenured will simply accept a pay raise and continue their work.