What is the difference between 秀才, 进士, 举人, and 甲第?
It's just like you attend the exam held in small village, then you passed it and became one of the top students. We call you  秀才 at that time.

After that, you are going to atten the exam held in a province scale, if you passed it also, then you became 举人. 

Only those 举人 can attend the national exam, you passed it so you became 贡士.

Only then, you can attend the exam held by the emperor and if you succeed, you became 进士. And the top one is called 状元, the second one is called 榜眼, and the thrid one is called 探花.

甲第 means the　[top successful candidate in an examination].