（よ）うとするtry to; be about to
You set about doing something: you make the attempt, or you are just about to. It doesn't say you managed, and what follows is often "but". （よ）うとしない is about someone who makes no move to do it.
tried to
I tried to open the door, but it was locked and wouldn't open.
about to
Just as I was about to speak to her, her phone rang.
won't
However many times my mother urges him, my grandfather won't go to the hospital.
How to form it
書く → 書こうとする 食べる → 食べようとする する → しようとする 来る → 来ようとする
The （よ）う form, then とする. Godan verbs (う-verbs) change the last kana to its お sound and add う; ichidan verbs (る-verbs) change る to よう. する then changes as usual: とした, としている, としない.
Trying to
Whether the attempt worked is left open, and in the past the next words usually say it didn't.
I tried to get up, but I couldn't.
I tried to apologize, but I was so nervous that no words came out.
My little brother is trying to tie his shoelaces by himself.
ようとしている: in the middle of the attempt.
Three kinds of "try"
English says "try" for all three. Japanese keeps them apart.
I tried natto.
てみる: you ate some, to see what it was like.
I tried to eat natto.
ようとする: you made the attempt, and may not have managed a mouthful.
I try to eat natto.
ようにする: you make a point of it, as a habit.
ようとする reports an attempt. It doesn't announce one, so it isn't for what you are going to do. A plan is ようと思います. A promise to make the effort, "I'll try to", is ようにします.
I'll try to get up early tomorrow.
Not said: 明日は
(I'll try to get up early tomorrow.)
About to
With とき or たら, it is the moment you went to do something and something else cut in.
Just as I was about to close the window, the cat came back in from outside.
It is also said of things with no will of their own, and then it sounds like storytelling. In everyday talk people say もうすぐ or るところだ.
When I got to the station, the train was just about to leave.
Another year is about to end.
ようとしない: won't
Someone could do it, you think they should, and they make no move to. It is mostly said about other people, as a complaint or a worry; plain 行かない would only state the fact. ようともしない is stronger: "won't even".
The child must be sleepy, but won't stop playing the game.
Even when she was asked why, she made no attempt to answer.
His fever has lasted three days, and my father still won't even go to the hospital.
Close neighbours
ようにするN4
Making a point of it, often with できるだけ.
For my health, I try to use the stairs whenever I can.
（よ）うと思うN4
The plan is still in your head. With とする you have made a move.
I think I'll be heading home now.
るところだN4
Plain "about to": the next thing you will do, with nothing cutting in.
I'm just about to have dinner.
Check yourself
1Did the writer read the letter?
手紙を
Choose an answer to see why.
2"I'll try to come early tomorrow."
明日は
Choose an answer to see why.
3What does this say about him?
兄は
Choose an answer to see why.
More detail
ようとしても
"However hard I try": what follows is that it can't be done.
However hard I try to forget, I can't.
More sentences
We tried to move the heavy desk, but even the two of us couldn't shift it.
I tried to enter the password, but the screen froze.
The baby is trying to use a spoon without help.
The child who fell is trying to stand up without crying.
However many times the clerk explains, the customer won't read the notice.
Even when there is something the new employee doesn't understand, they make no move to ask a senior colleague.
However many times we proposed talks, the other side wouldn't even come to the table.