Most courses have enough space for all who want to enroll, but there are a few that do tend to fill up quickly. The "tiered" registration process, where you sign up for only 12 credits on the first day of registration, is intended to give everyone a fair shot when resources are limited.
Some courses require permission numbers, which are usually available from the instructor or his/her assistant. If you have a permission number for a course, don't use it during your first registration slot because you will be able to use your permission number to register later. Instead, use your first registration slot to register for other courses you want to take that may have limits on enrollment, but that don't control entry by using permission numbers.
Beginning on the first day of each registration period, you can register for up to 12 credits. Once drop-add begins until it ends in the second weeks of the next semester, you can register for additional credits (remember to get permission for an overload if you want to register for more than 15 credits in a semester).
For incoming students registering during orientation week: If you don't get into every course you want, remember that you'll be able to pre-register for your spring courses along with second year students; and, next year you'll get priority access to fall courses because you will be pre-registering for those in March.
Returning students must wait until the end of orientation week to register for more than 12 credits in the fall semester. Student Services will let you know the dates on which you can register for various numbers of units for each semester's registration.