Kinds of Links
Link earning is still very much the foundation of SEO and is likely to not go out of style anytime soon. It's an indication to the search engines that you have good content which is worth linking to, a virtual thumbs up from another webmaster from ideally an authority site. To reward you for receiving your link from that other site, search engines give you a better rank, giving your site more visibility in the search engines. Link earning is far more effective (not to mention completely safe) compared to link building in which the webmaster or marketer creates links themselves through networks or link building softwares. These methods almost always result in at best a waste of time but at worst a drop in ranking or penalties coming down from Google. Google has gotten far more perceptive at sniffing out trails between two websites if the links have been manufactured and makes it a point to not reward these websites because there is no legitimate vouching for another site, it's just gaming of the system.
Now that I've given a brief overview of link earning versus link building, let's talk about the nitty gritty with kinds of links.
Internal Vs. External - Internal links are those which point to the same domain they're linking from. It's a wise idea to have a lot of internal links on every single page of your website within the content you're creating as this encourages people to stay on your website which in turn says to search engines that you have a website worth spending more time on, resulting in better rankings for that site. External links are those which point to a different website than they're linking from. While you can use internal links to same effect to pass link juice (more on this in a moment) around on a single site, external links are those which truly pass influence from one website to another.
DoFollow Vs. NoFollow - A link can either be made to be dofollow no follow. Both are live active links which if clicked will take you from one page to another. The difference between the two is that dofollow links pass "link juice", or in other words they pass influence from the linking from site to the linked to site. It's the influence which Google and other search engines recognize and reward with a bump in the SERPs for the linked to site. It's measured in an unquantifiable metric which we loosely refer to as link juice. If you want to link to another site but don't want to pass on link juice (such as with affiliate links), the nofollow tag is appropriate. Generally you don't have to worry about making nofollow links.
Sitewide - Sitewide links are those which appear on every single page of a website. Menu links are an example of this, and the vast majority of the time these links are internal, used to aide visitors of the site in navigation.
Now let's talk about the different link patterns possible. It should be made clear that you shouldn't be interested in any of these because once again you want to EARN links, not build them.
Reciprical - A reciprical link is a type of two way link between two websites which link to one another. It used to be a fairly common practice in the early days before search engines got wise to it as a way of gaming the system and jumping in the rankings without offering any real content. Now it's a waste of time.
Three/Four Way Links - Webmasters used to try to get smart and rather than making an obvious reciprical link, they'd hook up with two or more other webmasters to create a massive chain. The idea is that no two sites are linking to each other, so there's no obvious reciprical links but link juice is still being passed. Search engines got wise to this over a decade ago, so don't waste your time.
What is the Most Valuable Link?
Arguably the most common question I get regarding links is what is the most valuable link which you can get for your site. The answer will always be a one way link coming from another site, and an authoritative site at that which has a lot of influence from other sites linking to it. One way links always look better because there's no give and take. The perception to search engines with one way links is that if your content is good enough, it will earn a link. The other webmaster is getting good content in return for their visitors.
Remember to create good content based around targeted yet less searched for and therefore less competition based keywords, promote it via social networking, and you'll rank for them. Then, as your website grows and receives more and more links, you can begin targeting the higher volume keywords in time and the entire thing will begin to snowball in the direction you want.