For the purpose of discussing how we think, I'm going to use the word "intuition" to describe arriving at conclusions or making decisions that are pretty much instantaneous, as opposed to "reasoning" that proceeds in multiples steps. This is common, everyday thinking, and is not intended to imply any supernatural process like "I have an intuitive feeling that somebody is watching me."
We acquire intuition through experience
As we go through life, we have lots of experiences. The first time we encounter a situation we might not know what to do or what is going on, but as we gain experience with similar situations, we learn what actions work the best and what judgments are appropriate. If we have directions to someone's house, the first time we go there we might make some wrong turns and have to consult the map. If we have been there many times, we have learned to recognize the proper places to turn, so going there doesn't require any thinking. This is what I mean if I say we are operation on intuition. We simply recognize patterns (like the street corner where we need to turn) and make the response that has been most appropriate in the past. No reasoning is required. If there is road construction and we cannot use our intuitive response, then we need to stop and think and do some reasoning about what is a suitable alternative route to the destination.
Much of our behavior works on the basis of intuitive responses to familiar situations. If we meet someone we know named Fred, we are likely to say "Hi, Fred" without having to think it over. If we want to catch a ball thrown to us, we move our bodies and hands to the appropriate position without any thought, assuming we have had a reasonable amount of experience catching a similar ball. If we are eating a meal, we know when and how to use a fork if we have often used one before. If a bug lands on my arm, I'll usually swat it without any thought, unless I know there are bees around and I should find a safer way of getting it to go away.
Beliefs also often depend on intuitive knowledge. Sometimes this is essentially the same as behaving intuitively. Deciding that it is safe to cross the street can be seen as an action we take when there is no traffic, but it also involves the belief that there is no traffic that is likely to hit us.
We might believe the tree in front of us is a maple tree because we have seen many maple trees before and recognize the shape of the tree and the shape of its leaves. We might believe that the batter was out at first base because we saw the infielder throw the ball to the first baseman before the batter got there. We might believe a dog is not a threat because it is small and its owner has it on a leash, or that a piece of fruit should be thrown away because it looks spoiled. These are common sorts of beliefs that are based on recognizing a pattern rather than some sort of reasoning process. These are also relatively uncontroversial.
Strengths and weaknesses of intuition
A lot of the point of responsible thinking is to recognize when intuitive beliefs might be faulty. Our intuition might tell us that important, sophisticated people drive a certain make of car because of what we saw in a television commercial. We might believe that a politician is very virtuous because he talks a lot about how great our country is and always surrounds himself with flags, and we intuitively associate these things with virtue. We might form the intuitive belief that people from a certain ethnic group are stupid because they don't wear stylish clothes. Our intuition is likely to tell us that eating chicken is dangerous because we saw a news story that some people got sick from eating tainted chicken. These are cases where our intuition could easily be faulty, and sometimes, as in the case of the commercial or the politician, people are deliberately trying to manipulate our intuition.
For the most part, however, intuition works very well, and we could never get through life without making lots of immediate judgments about the things around us and how we should respond to them. It would be impractical to replace all intuitive judgments by careful logical reasoning, and most of the time it wouldn't give us any better answers than we get intuitively.
Instead what we can do is develop one more intuitive skill - that of recognizing when a situation is one which commonly fools people, such as when there is an emotional difference of opinion, or someone has something to gain by persuading us of something or when an extremely rare but sensational mishap occurs. It is in these situations that it pays to stop and try to employ careful reasoning to reduce the chances of adopting a false belief.