Graduate student Jon Armel giving a lecture
If and when you apply for academic jobs, you will likely be expected to have some experience teaching a class on your own. Students who have advanced to candidacy are eligible to be considered for an Associate Instructorship. Associate Instructors are grad students who are placed in charge of teaching an entire class, usually Math 3C/4C (Pre-calculus) or some other lower division class. Associate Instructors supervise TAs, write up a course syllabus, give lectures, midterms and final.
The Departmental Council has mandated that students who get Associate Instructorships must have very good TA evaluations to be considered. They are also looking for grad students who have been TAs for the elementary math classes like Math 4 or Math 10, so you should get some good evaluations from those courses.
The actual procedure is to notify the Graduate Vice Chair that you are interested once you have advanced to candidacy. The Graduate Vice Chair maintains a waiting list. Those with weak TA evaluations are often not considered because there are so many grad students who have good evaluations. The number of openings is limited. Generally there are 4 openings in the Fall quarter, 3 or less in the Winter quarter and maybe 2 in the Spring. There are also opportunities in the summer to teach summer school in either one of the two 5 week summer sessions (one in July and one in August). For those looking to teach in the summer, more information can be found here.
Take a look at the University's policy's regarding teaching at UCSD to learn more about the rules and regulations Associate Instructor's must follow: