this (person/thing)

In the example above, the speaker uses "this" instead of "a". There's a small difference between the two. When you say "this ___" it sounds like you're talking about one specific thing. "A ___" is more general. 

You use "this ___" instead of "a" or "the" when you're introducing something specific that you're going to talk about. It's usually used at the beginning of a story or explanation. For example:

Amy and I were sitting there talking and this guy I know walked by.

I've had this song stuck in my head for two days!

Today I was talking with this girl at work who just got back from maternity leave.

In all of these examples, the speaker will continue by telling a story about that topic.

People only use "this ___" in this way in casual spoken English.

an incredible (thing)

"Incredible" means "amazing" or "really good". Some things that people often describe as "incredible" include:

an incredible story

have an incredible experience

get an incredible opportunity

have an incredible amount of something

a documentary

A "documentary" is a documentary film. This is a film that's based on real events instead of a made-up story.

the other day

"The other day" means "a few days ago". You use it to talk about a day that was 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.

Here's one example that you can use when writing to someone you met last week:

It was great meeting you the other day.

the state of (something)

This phrase means "the situation" or "the way it is". For the example sentence at top, it means "The way that public schools are."

Technically, the phrase "the state of ___" is neutral. You should be able to talk about either positive or negative things with it. However, people mostly talk about "the state" of negative things.

the public school system

Most schools are either private schools, which parents pay for, or public schools. Public schools are supported by money from the government.

It's common to talk about the "school system", which is the overall group of schools in a city, a state, or a country. For example:

The Franklin City school system is doing quite well these days.

So a "public school system" is a collection of schools that are all operated by the government. In the example at top, the speaker is talking about all schools in the country.

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.