I'll explain -아/어 and -고, two of the most common conjugation forms of verbs and adjectives.

1. -아/어 = the most heavily used one. It is like "and" in the sense of tightly connecting a verb with another verb or a phrase, as in "go (and) change your clothes". 

E.g: 빼 놓다 (빼어 놓다), 해 주다 (= 하여 주다), 없애 버리다 (= 없애어 버리다), 치워 버리다 (= 치우어 버리다).

(The parentheses contain the intermediate forms before it shortens to their final, smooth forms)

Each of these combinations is almost like one word because the two parts sort of fuse to create a somewhat different meaning.

2. -고 = like "and", but not as tight as -아/어 in its connection.  It has two major senses.

A: Enumerates two verbs or phrases.  너는 너고 나는 나다 (You are you, and I am me = We are all individuals).

B: Sequences two actions, keeping their distinct meanings.  밥을 먹고 밖으로 나갔다 (I ate, and then went out).

Some usage examples:

1. 책꽂이에서 책을 빼어(서) [or 빼, 빼서] 펴 보았다 = I removed a book from the bookcase and opened it to have a look.

2. 영철이는 축구를 잘하고 난 농구를 잘한다 = 영철's good at soccer, and me at basketball (-고 A sense)

3 영철이는 몸이 약하니까 빼고 우리 셋이 하자 = Let's do it between the three of use, excluding 영철 who's too weak. (-고 B sense).

Sometimes the distinction between these three senses are not so sharp, but you should still keep them in mind as a guide.

You can't just remember a few simple rules to know how to use these conjugations.  Like in any language, you'll need to practice with loads and loads of usage examples.

It depends on the sense "remove" is used in.

Here are the related words.

- 빼다 = take/pull out (from a group or a place), remove, exclude, subtract.

- 제외하다 = exclude.

- 없애다 = remove; get rid of.

- 제거하다 = remove, get rid of (similar to 없애다, but more formal).

- 치우다 = move/remove (out of the place); tidy up.

  

For example:

- I removed/pulled out the nail from the wall = 벽의 못을 빼/뽑아 버렸다.

- Who removed the vase from the table? = 테이블의 꽃병(을) 누가 치웠어?

- They removed him from the board = 이사회에서 그를 빼(어)/제외해 버렸다.

- Removed all irregularities from the work procedure = 작업 과정의 모든 불규칙성읠 없앴다/제거했다.