I was wondering what the difference is between sponsored links and the other links that come up when you search on Google. I was told you only have to pay for the sponsored links if you're using Google Adwords. Everything else I was told is free? But how can you have your website come up under the regular links as opposed to the sponsored ads/links?

Basically, getting your site to come up in the 'regular' listings is where Search Engine Optimization comes in. The object of SEO is moving your site up in the rankings for the keywords that your prospects would use to search for what you sell/provide.

There are various techniques to SEO, some very complex, some not so difficult. It's a competitive area, though, as those top 10 or 20 rankings are very valuable.

You can find a lot of sources online about SEO, just search under 'Seach Engine Optimization tutorials' or something like that. You also might want to look into hiring a professional.