instructor, assistant, professor

They differ from country to country. I presume you're referring to the US?

In the US, the ranks (from highest to lowest) are typically Professor, Associate Professor, Assistant Professor, and Adjunct Professor/Instructor. An assistant (e.g., teaching or research assistant) is typically not considered a faculty member though he or she might teach classes (usually an assistant is a student working in a field related to the degree he or she is pursuing; for example, I was a teaching assistant in the department I earned my master's from--I worked 20 hours/week, they paid my tuition and gave me a stipend to live on). An adjunct professor/instructor is signed to a limited contract (either signed on for a semester or an academic year, but their employment from term to term or year to year is not guaranteed); typically, an adjunct's pay and benefits really suck, but an adjunct need only have a master's degree (I did this for two semesters--it's a great part-time job, but you wouldn't want to make a career out of being an adjunct). An assistant professor has, with a few exceptions, a Ph.D. and is on the first rung of permanent academic employment; an assistant professorship is what's called a "tenure-tract" position--if the new professor does his or her job adequately and publishes the requisite number of articles/studies/etc., then he or she will be granted tenure and promoted to associate professor after a few years; otherwise, an assistant professor can be fired pretty easily if he or she screws up somehow (think of it as being a probationary employee). An associate professor has tenure (this means it's more difficult to fire the person) but is still required to publish a certain amount of material on a regular basis. The only real difference between a professor and an associate professor is that the professor has been a working and publishing academician longer than the associate professor.

The specific criteria for advancing from assistant professor to associate professor to professor varies from institution to institution, but the order of prestige is pretty standard.

That's a good summary. The only thing I would add is that while assistant professorships are usually tenure-track, what is required to actually get tenure varies by institution. "Adequate" is often not enough to get tenure at someplace like Princeton. The top institutions have something like a 50% tenure rate -- an assistant professorship there can amount to a glorified postdoctoral position.

The time at assistant professor (before receiving tenure) also varies, with a few years usually on the short side of things (tenure is often granted after a longer period.)

Note, I'm sure all of this also varies by field, it's what I know from being married to a new assistant professor in a scientific field.