If you're referring to "Meta Tag Keywords", Google does not use those anymore. Quite a few years ago they decided to start choosing the keywords themselves, which gets based off of the content on your website.
The keywords are chosen based on Keyword Density which is the amount of times the keyword/phrase is used on your webpage. It's a bit more complex than that though... because if there is too much Keyword Density then it can have a negative impact because they believe the site won't look natural. Also, Google looks to the Title tag and H1 tags on your webpage as very important so you will want to make sure your keyword/phrase is in there too.
As for your questions..
1. The simple answer is that keywords are what people will search to find your website.
2. You implement them in your title and content of your webpage in a natural way.
3. See answer 2. However, there are other factors. Like you might be competing for a really competitive keyword/phrase and not appear in the search results. You can get penalized for a bunch of different things. Like keyword stuffing (unnaturally placing keywords in your content). Getting links to your website from other websites can help you with improving the rankings of your keyword/phrase. This is how you can beat the competition on Google as long as your onpage stuff is ok. But sometimes the links can hurt your rankings too if they are too spammy.
Edit: Also, you can run into issues with duplicate content. Like say if you just copied your article from another website. Google can penalize your rankings (and/or make them non-existant). There are tons of tools out there to run on your articles to make sure they're unique, try Googling copyscape.
I would suggest that you start off your site targeting non-competitive keywords. They'll only bring in a small amount of traffic usually, but it will help you get over a big learning hurdle. Target some 4+ keyword phrase and make a unique article based around it. Just make sure you get all the onpage stuff taken care of and ask questions here if you need to. Then see how well you rank and go from there with backlinking if you need to. Then set your goals higher
P.S. It can take time for new pages to show on Google. Who knows, maybe your article is in great shape, and you just need to wait a bit longer for it to get indexed onto Google. You can also submit sitemaps to speed it up too.
