There is a lot of gray area here but the simple answer to your question is yes, the more websites (not people) that link to you, the more popular you are perceived by the search engines and yes, the higher you will go on the natural or organic SERPs (this is part of search engine optimization).
While links don't have to be relevant to you, it's better that they are at least in the same genre as yours. What I mean by this is that if you sell tires, a link from Ford.com will count more than a link from a site like WebsiteMagazine.com - even though we are both respected leaders in our fields.
People do pay to get links to their sites. This is usually achieved through direct purchase from another website or through services such as Text-Link-Ads which help publishers (website owners) sell space on their site and help advertisers buy space on the publishers site. I would personally not advise buying, renting or leasing links from another website - the best links are earned links.