These are some really challenging words to learn to use in English.
A is used to show that you are talking about a single object, it's another way to say "one". For example "I'm going to buy a car" / "I'm going to buy one car", both sentences mean the same, but using a is more natural. (Remember that if the word after a starts with a vowel, you have to change to an, just to make things complicated. Example: "I want to eat a banana", "I want to eat *an* apple.")
Or shows the choice between two or more things. For example: "Are you going to the store or the movies?" it shows a choice. "Should I buy one or two pencils?"
In shows that something is inside something else "There is water in the cup", but it also shows time, "In two days I'll drive to the city." So the first one says that the cup contains water, and the second says that the speaker will drive to the city two days from today.
On is like in, but it shows that one object is touching the outside of another object. "The pencil is on the table", like in this picture: https://vignette.wikia.nocookie.net/battlefordreamisland/images/6/6e/Pen_and_Pencil_on_the_table.png That's the most common usage, but it can also indicate time, "On Monday I go to work" means that the speaker will be going to work on Monday.
Hope this helps! Good luck :)