Being indexed means that Google has crawled your site and added it to their database. It can be found - even if very rarely or with great difficulty - in SERPs. An easy way to check if your site has been indexed is to do a site: search (that's site with a colon after it). Put your URL after site: in Google's search bar and hit enter. Only results for that URL should show up. This way, you can get an idea of how much of your site has been indexed.
Ranking is where your site shows up in SERPs. There are a lot of different ranking factors. Just because you don't show up on the first page for a certain search doesn't mean you aren't indexed - it could mean that you're indexed but your site doesn't have enough authority to outrank the sites ahead of it.
Many newbies, probably after reading SEO blogs of questionable value, seem to jump right into link building and off-page SEO immediately. That can be dangerous and is probably the wrong way to think about things.
So let's be clear - being indexed and ranking are two very, very, very different things. Not ranking, not being indexed, losing rankings, and being dropped from Google's index are four separate problems that often require unique solutions. Don't make the mistake of confusing indexing and ranking!