Obtain the transfer equivalency chart and your new school's degree plan for your major. Most universities publish a transfer equivalency chart naming common course titles and their in-house equivalent. Many even post these online. If you are transferring from a nearby university, your new school may have a chart specifically for that university. Degree plans are similarly available online.
Audit your own transcript using your degree plan, transfer equivalency chart, and course description book. Some courses will be automatic transfers, while others may be a judgment call. For courses that are not directly transferable, get a course description book for both schools, and compare the descriptions. If you think it should transfer, make a note of what course it should transfer as.
Submit transcripts for transfer. After you have been admitted to the university, your transcript should be forwarded to the registrar for a transfer evaluation. Find out if the admissions office does that automatically, or if you must initiate the process. Once the registrar's office has your transcript, a counselor will go through it and accept what he thinks is suitable.
Some courses will transfer as direct courses, some will transfer as electives, and others will not transfer at all. In general, universities will not accept for credit any grade below a "C." They will also not accept any remedial coursework, and certainly not any incomplete courses. The most common problem is a course being transferred as an elective, rather than fulfilling a requirement.
Appeal any decisions you disagree with. You may find their audit to be on their conservative side. Ask your transfer auditor how the university's appeals system works. It may consist of filling out a form, submitting the page from the other school's course description book, and having it signed off by a department faculty member.
Prevail through the red tape. Universities can be bureaucratic and inefficient. However, you must respect the process. Fill out the forms. Get the signatures. Stand in the lines. Leave the voicemails. Above all, remember that you are getting something done, even if it does not seem that way.