W.W.W. and how it works?

I know how to turn on the computer, find my Internet service provider and that is it. So I know nothing about how the Web works.
Question: If I have a website, and to find it, you would have to go to www.MyFish.Com. Not a real one, by the way. Is there any way to arrange the web when you do a search? If I typed in fish, could I do anything so my website comes up first, second or third? And why do some sites don't come up at all? The reason I ask is I started going to a forum that has been up for about a year. The person who has it started it, one month after he started it, got sick, and nobody knew. I just found out he has cancer. He didn't assign a moderator to help. I just happen to go there and see no one was helping this people out with their animals. The last post was in Nov from the main people who use to post there. People would ask a question and never get answered so they went I started in Feb, answering questions. This month he was feeling better enough to post. He asked me to help him advertise his web site to get more people on it. So I was thinking if he could do something so his forum comes up to the top and not down the bottom.
As of now, he has 4 people on it and that includes me. I really don't have any idea how you advertise his site. So if you have any ideas about that, let me know.
He is 17, lives in Croatia and has problems with his english, so
I told him I would help. As he has this on his forum,
"Please, make a search so that I will have some profit and click on links below. It only takes 5-10 seconds. Thanks!"
I figured this would be business related. So I could post it here. As I think he needs the money. I have to work on him to change the wording on it but it makes me smile when I see it.

Webmaster World

That link will contain a treasure trove of information so you can locate viable solutions that will promote your site. It will also keep you away from the scams that are out there. I've successfully promoted over 100 sites successfully for clients that last 3 years using information similiar to what you will find on that site.

Be prepared for a lot of reading and work. There is no easy "magic pill" that makes it happen (except to have MicroSoft's marketing budget ). Just hours of work.

Hope that helps,

Thank you very much. I can see I am going to be doing for the next week. Reading and more reading.