Are backlinks harmful?

Avoid them at all costs.

Bulk building links in this way is the quickest way to get your site de-listed despite their claims. Bear in mind they want you to pay them money so they will tell you what you want to hear. 

As for what links are good, well nothing there has changed, users who link to you that are naturally gained are the best ones to get. Better ones are from high authority sites but they don't tend to link to other sites at your call. As an example, let's say the BBC did an article about a topic and posted an unprompted link to you. That would be the perfect link and gain you a lot of creditability. 

In addition to what Chedders as provided you....

Links with the 'No Follow' tag will not hurt your site.  You should be building backlihks for traffic and not ranking.  

Google now uses the tags, "no follow, sponsered, and ugc", as hints and may in fact count them.  The links will not impact your seo in a negative way.   I look at it this way, they are mentions and mentions do carry weight.  How much weight is up for conjecture.

As Chedders said, stay away from bought / paid links, they just cause trouble in the long run.