Think of vs. Think about

Remember
Think of
In this sense think of usually comes with can.
- I can’t think of her name at the moment.
- I can think of at least three occasions when he arrived late.
- I can’t anyone who could speak English.
- Nobody could think of anything to say.

Think of/about
To recall a thought or an image to mind when sth happens .
- She thought of/about her childhood when she saw the movie.
- I thought of/about you when they said they wanted someone who could speak English.

Imagine
Think of
Imagine
- I have never thought of becoming an actor.
Notice how the meaning of the sentence changes when we replace think about with think of.
- With think of, it means that you are not an actor now, and you had never had thoughts regarding taking this career path. If you were to say "I had never thought about becoming actor," I jump to the conclusion that right now you are an actor, but you had previously not considered this as a career choice.

To have a particular opinion about someone /something.
Think about/of
- What do you think of /about his suggestions?
- What does he think of /about the new situation?

Note the following sentences:
- I’m looking for a partner for my new business. What do you think about John?
- I’m going to give one of my staff a promotion. What do you think about Tom?

If you are asking for an opinion about what someone would be like in a particular role, you may use think about.
If someone is asking you for an opinion about how a person is like when for example you meet them, you usually use think of:
- What do you think of your teacher?

To direct your thoughts to sth/sb , to occupy your mind with sth/sb.
Think about/of
- Do you ever think about/of your childhood?
- I can’t stop thinking about/of her.
- Did you think of/about me at all when you were with her?

To mean, intend:
Think of
- When I made those remarks, I was not thinking about anybody in particular.
-I was thinking of you when I complained about people who gossip.

Consider
Think of sth
1- As is just used with think of and not think about in this sense,To Consider someone as:
- Do you think of me as your lover? (Do you see me/consider me to be your lover)
- Do you think of him as your friend?
- In China bats are thought of as being very lucky.
- He thinks of it as his home.

Note that think of someone as is synonymous with consider, regard.
- In this country many people consider/regard/think of him as a hero.

Think about /of someone or something
2-Think of/about sb and show consideration for them and their needs. To have concern. Usually used with think of:

- You never think of anybody but yourself.
- Don’t you ever think about/of other people?
- We have the interest of 50000 customers to think of.
- You don’t have to think about Hugh and me.
- Think first of the ones you love.
- A man must think first of his family.

3- To consider sb/sth that you might possibly do or plan in the future:

- Is he still thinking of/about going away to Italy for a month?
- Who is she thinking of/about marrying?
- That’s why I was thinking of/about getting out.
- She’s thinking of /about changing her job.

Notice that:
Think about means to weigh or investigate an idea before doing it. To investigate the problems related to it.
For example if you are thinking about investing in a project, it indicates that you are investigating about the profit and loss of doing this.
Think of implies that you are planning to do a work or you have the idea of doing a work.

Think of
To consider the suitability.
-We thought of her of Miss the world.
- Many people think of her as president.

Think of someone/something.
To consider someone/something, to take someone/something into consideration:
-There are so many things to think of when buying a house.
-You can’t expect me to think of everything.

To produce an idea in your mind using intelligence and imagination:
Think of
- Roman civil engineers could think of solutions to every problem.
- Can anybody think of a way to raise money?
- He can’t think of any reason/excuse for not going.
- No one before had thought of bifocal glasses.
- I need to think of a solution to the problem= It means that I need to find an answer. But, if I say "I need to think about the solution to this problem," I mean that a solution has already been proposed and I need to consider whether or not it will work/is practical/etc.

To think about something a lot especially you are worried or unhappy about it.
Think about
- Rebbecca is still very upset, and she can’t stop thinking about the accident.
- I wish she would stop thinking about work, she never relaxes.
- Just dismiss those thoughts from your mind, they are crazy and not worth thinking.

To ruminate, consider a matter carefully:
Think about
- He thought about the problem of recession hard.
- Tom thought about his past mistakes and how to compensate them in the future hard and carefully

Notice that:
1- Sometimes,think of or think about is the matter of style:

- I was just thinking of the wonderful trip we had. - it gives a thrilling moment, a person enjoyed a trip so much that it was beyond imaginable.
- I was just thinking about the wonderful trip we had. - a person recollects events from the trip with no special emotional attitude.

2- Note the comparison:
- I don’t think much of the new president.
- It indicates that you think he’s bad, useless or generally you don’t approve of him.
- I don’t think much about the new president.
It implies that you don’t spend any time thinking about the new president. it relates to the action of thinking.

3- I am thinking well about my trip = quite carefully. "Well" here means carefully, or thoroughly.

- I think well of the idea = I think it's a good idea. "Well" here indicates your approval of the idea.

References:

English Forums.com
English-test.net.
Free dictionary.com
Cambridge Advanced Learner's Dictionary
Longman Advanced American Dictionary