
Do you want to increase your pagerank? Do you want to know how you can  increase your pagerank? Increasing your pagerank is much easier than  understanding its algorithm, my friend. Read on, I will explain it in  the simplest way I can.
As an introduction (just in case), pagerank represents the 
importance  of a website in the vast internet, where a higher pagerank means better  importance. Pagerank, which are scaled from 0 to 10, becomes  interesting when involves traffic, those with more ranks will win more  traffic from search engines.
Before starting, you should know that pagerank is updated once in 
3 months. Therefore, you should not expect any changes soon after you start working on it. Meanwhile, you can try to know it first (
What is PageRank), then try understand a bit of its pattern (
Algorithm of PageRank).
I know you are too busy for that, so in 
summary of its algorithm,  dofollow inbound links (backlinks) are the most critical aspect in  determining your pagerank. Dofollow means that your link will be crawled  by the bots / spiders of search engines, nofollow is the contrary, and  both are set using anchor text attributes. Also, more ‘important’ (high  ranked) website votes more. If all your pages are linked (dofollow) each  other perfectly, you already get a pagerank of 1. Those are the main  points.
So, here are some methods that can build your pagerank, starting from the free ones:
1. Internal DoFollow LinkingGood dofollow linking (internal site navigation) inside your website  will be helpful in increasing your pagerank, though it can only reach a  pagerank of 1 without external links. It can be easily done using an 
archive link / widget that is located in your sidebar, or a 
sitemap.  It will render all your posts / pages connected each other handsomely,  since your sidebar / sitemap goes to all your posts / pages.
2. Natural DoFollow LinkingWhen bloggers write something, they usually put some external links as  an additional information for their readers. The links inside posts are  usually dofollow, unless the bloggers specifically and manually defines  nofollow attributes.
Here is how you can apply it: first you need to write 
good contents.  If then people read and get impressed, there is a good chance that they  will put your links in their own website / articles without asked (as  additional resources). It gains you more dofollow inbound links, gives  you more votes, and you grows your pagerank.
More great (not just good) contents means more chance of getting natural dofollow backlinks.
3. Blog CarnivalsBlog carnivals are blog posts containing submitted links of articles in a certain topics. 
BlogCarnival.com  is a gateway for writers / bloggers and carnival hosts to interact.  Writers submit their articles to the hosts through the website.  Afterward, the hosts choose the articles they like and provide the links  inside the carnivals, which are located on the hosts' websites.
Carnival hosts usually give dofollow links, because it is too much work  to add nofollow attributes to each of the links, especially when there  are lots of submission. If your articles are accepted, your dofollow  backlinks increase in number. The hosts vote for your pagerank, so it  shall rises.
4. DoFollow Link ExchangeLink exchange between sites / blogs is a common thing in the blogosphere  / internet. If you have some friends that have websites or blogs, you  can ask for dofollow link exchanges. When you can get more of this  exchanges, you will earn more votes. Thus your pagerank will increase,  very simple.
Getting some dofollow link exchanges with 
quality sites (higher ranked and relevant) are even better.
5. DoFollow ReviewsAsking bloggers to write reviews / advertorial about your site (with  your dofollow backlinks inside them) will also help building your  pagerank. Some reviewing services gives you dofollow links by default,  while in some others, you need to manually state that you require  dofollow links in the reviews.
You will need to spend some funds for this method, which is best for products and services.
6. DoFollow Text Link AdsThere are alot of websites that provide services in text link ads. Some  of them provide dofollow text link ads, which can help you in bulding  your pagerank. These type of ads will make the websites (where they are  located) vote for your site and increase your pagerank.
Text link ads are usually not free.
Discussion and Questions - Answers:1. How high can my pagerank grow?Common bloggers /online business persons can reach 2-5 of pagerank with  some efforts. Super successful and famous bloggers (like Darren Rowse of  Problogger.net and Brian Clark of Copyblogger.com) can get 6-7 in  scale. Social medias, like Twitter and Facebook, can obtain 8-9 in  pagerank. Search engines, like Yahoo and Google, can get the ultimate  9-10.
If a blogger wants a pagerank of 8, he's dreaming.
2. Can leaving comments in other blogs / websites increase my pagerank?Only if the blogs are dofollow, which means they allow search engine  bots/ spiders to crawl your links inside the comments sections. Most  blogs in the net are nofollow in their comments section to avoid abuse  and spam. Leaving comments will increase your link popularity, but not  pagerank. You need dofollow backlinks to increase your pagerank.
3. Can social medias increase my pagerank?No, they cannot. All social medias automatically add nofollow attributes  to all links inside them. Same with commenting, it will gain your link  popularity but not pagerank.
4. Black Hat pagerank building may work, why don't you mention it here?Search engines actively fight black hat pagerank building (like keywords  spamming, hidden link, etc), because they want to ensure that high  ranked sites are really valuable to online society, not just because  they can hack things using their techie knowledge.
It's like getting money, you can get money from providing satisfying  services to others, or by robbing or stealing from others. Both give you  money, just which methods you want to use. Natural and black hat  methods can increase pagerank, your purpose defines which one you will  choose.
Besides, using black hat methods can result in ban or penalty from search engines.
Note: If you want to know how I earned a PR 5 on my new blog, I've written my unique custom methods here: How to Increase Pagerank.