I feel like (clause)

You use "I feel like ___" to express your opinions. When you state your opinions this way, it sounds more personal than when you say "I think ___":

I feel like kids these days don't have the same sense of respect that they used to when I was growing up.

You can also use "I feel like ___" when you're not quite sure of something. For example, if you lost your credit card:

I feel like I might have left it at the bar I went to the other night.

the other (day/night)

"The other day" means "a few days ago". You use it to talk about a day that's about 2 to 10 days before today.

So you wouldn't use it if you're talking about something that happened yesterday, and you wouldn't use it for something that happened three weeks ago. You use "the other day" when you don't exactly remember what day something happened, or when it's not important to tell the exact day.

I was talking to somebody about that just the other day.

You can also say "the other night", "the other evening", etc.

"The other day" and "the other night" can be used in business or casual situations.

(something) might have (happened)

Use "might have" to talk about something that you're not sure about, but which would have already passed. For example:

A: Claire was in a bad mood.

B: Yeah, She might have been hungry or something.

In this example, the speakers are talking about Claire's mood in the past. But the second speaker isn't really sure if Claire was hungry or not.