You're eating dinner with some coworkers and someone mentions the topic of education. You watched a really good film about education last week. You want to talk about it, so you say this.
I saw this incredible documentary the other day on the state of the public school system.
In casual conversation, you use "this" instead of "a" when you want to start telling a story about something. Describing something as "this ___" shows that you're talking about a particular person or thing, not just any one. It also suggests that you're going to give more details on the topic. In the example above, it means that you're probably going to describe more details about the documentary. Some other examples of using "this" in this way include:
"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" is a documentary film. This is a film that's based on real events instead of a made-up story.
"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:
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.
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.
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