As a content writer you don't need to worry much about follow/no follow but here is a simple explanation of the concept and a best course of action.

What is a Link or Backlink?

A link is any text that can be clicked by the reader and leads to another webpage.  Links make it easy for the reader to go to the referenced website, and they also get followed by search engine bots to discover other webpages and rank them by importance.

What is PageRank (also called PR)?

Google search was originally largely based on a concept called PageRank. According to the PageRank system, links are like votes. Basically the more links pointed a site from other sites, the higher the PageRank.  CNN.com has more links/votes created by website owners then whatmydogdidlastweek.net does.  So the PageRank of CNN is higher and this is a factor in what Google returns in the search results.  PageRank runs from PR10 to PR0 and high PR is good.

The web is not a democracy and not all links are created equal. According to PageRank a link from a highly regarded site (with high PageRank) is more valuable than links from insignificant sites (with low PageRank). So if CNN or a big university links (say a PR9) to your website that is a bigger deal than if your mom's little vacation blog (PR0) links to your website.

Simple Page Rank ModelCredit: Wikipedia Creative Commons

This model from Wikepedia shows "Mathematical PageRanks (out of 100) for a simple network (PageRanks reported by Google are rescaled logarithmically). Page C has a higher PageRank than Page E, even though it has fewer links to it; the link it has is of a much higher value. A web surfer who chooses a random link on every page (but with 15% likelihood jumps to a random page on the whole web) is going to be on Page E for 8.1% of the time. (The 15% likelihood of jumping to an arbitrary page corresponds to a damping factor of 85%.) Without damping, all web surfers would eventually end up on Pages A, B, or C, and all other pages would have PageRank zero. Page A is assumed to link to all pages in the web, because it has no outgoing links."

What is Link Juice?

PageRank can be understood to flow like juice between sites which is why some people call this "link juice" or "Google Love". A backlink is just a term for a link elsewhere that links back to your site. One sites outbound link is another's backlink.

In a closed system with one owner (say a single large website) PageRank as conceived works great. The pages that the webmasters link to a lot get ranked highly in search results and unimportant pages that get few links are ranked at the bottom of search results. Unfortunately introducing the desire to profit changes things significantly.

Problems with PageRank

The big problem with link counting and link ranking based PageRank is that many people try to rig the system to make more money. They automatically would build thousands of links themselves (link spam). They created link farms with no useful content. They started selling links on high PR sites to the highest bidder. They started doing link exchanges between unrelated websites with dedicated links pages. None of these techniques added value to the web reader looking for good information.

Black Hat vs. White Hat

The whole SEO industry exists to get websites noticed by search engines using various ever changing techniques.  "Google approved" techniques are known as White Hat while doing bad things (in Google's eyes) is called being Black Hat. So putting good tags on your article to help the search engines is White Hat while using a bot to create a hundred thousand spam comments linking back to your website is Black Hat.  The colors harken back to the old western movies where the bad guys would almost always dress in black and the good guys often wore white.

Nofollow

Google keeps changing things up to fight efforts to manipulate the system by Black Hat SEO practitioners.  One of these efforts to fight Black Hat is the introduction of "no follow". All links used to be "follow" until the idea of "no follow" was invented in 2005. A "no follow" link is not supposed to pass along PageRank. 

Here is an explanation of nofollow from Wikipedia: 

In early 2005, Google implemented a new value, "nofollow", for the rel attribute of HTML link and anchor elements, so that website developers and bloggers can make links that Google will not consider for the purposes of PageRank—they are links that no longer constitute a "vote" in the PageRank system. The nofollow relationship was added in an attempt to help combat spamdexing.

As an example, people could previously create many message-board posts with links to their website to artificially inflate their PageRank. With the nofollow value, message-board administrators can modify their code to automatically insert "rel='nofollow'" to all hyperlinks in posts, thus preventing PageRank from being affected by those particular posts. This method of avoidance, however, also has various drawbacks, such as reducing the link value of legitimate comments.

Most blog comment links are now automatically set to nofollow but there are exceptions including blogs that run the commentluv plugin.  

Does "nofollow" Matter?

If you are a writer or blogger looking to create a few anchored backlinks to help your work get noticed, nofollow does not matter to you.  For most purposes a link is just a link but some links are better than others.  A natural link profile includes a variety of links, some from high PageRank sites and some from low PR sites. Some will be follow and some nofollow. 

What is Deep Linking?

Deep linking is sending links to content pages across your website (or specific articles) instead of just linking to the home page. This creates more authority for the site over all and provides additional benefits more fully explained in this article about deep linking.

Is Backlinking OK?

In a way, by building our own link we are trying to manipulate the search engines, but done properly on a small scale this is perfectly acceptable.  It is very reasonable for me to reference an Infobarrel article I wrote from a blog post I wrote.  I can share my articles with my friends on Facebook - Google expects that.  Linking a coin article from a coin forum is a good thing as is linking to my article from a Top List I write. These are all good reasonable ways to  build links and attract traffic from the search engines. Without links the web would not work at all, so many links are good.  

Taking things to extremes and spamming the world is, however, very not cool and will be punished by Google by sending your webpages to the bottom of the search results (or even deindexing the pages).