Amazon allows you to host giveaways of just about anything. You can give away your book (as an E-Book or Paperback) or things like new kindle devices or anything people might be interested in. It's incredibly easy to set them up, and there are some really cool features involved that authors can take advantage of to help drive sales.
Bonus for People Entering Giveaway
When you create a giveaway on Amazon, you get to choose a condition people have to meet to be able to enter your giveaway. This isn't Rafflecopter where someone might be able to get extra entries for subsequent tasks, but rather a single cost of entry. There is also a condition where you can't ask people to do another task to be able to enter your giveaway, but you're sharing a link so this shouldn't be a problem.
A benefit is that Amazon is constantly improving this and adding new ways to enter. It started with basically just twitter follow (where a user had to follow you on twitter to enter your giveaway) and now they have added a lot of other things like watching videos to be able to enter and Amazon Follow.
Amazon Follow is quite possibly the best method of entry you can use, because when you release a new book Amazon can contact readers about it on your behalf to remind them it is out. Having followers can introduce a viral effect to your sales that can payoff for years to come.
One of the good and bad sides of this is that it is Amazon controlling these lists: Amazon lets you send messages to people about book releases, but you can't spam people. Further, the emails come as recommendations from Amazon, which carries weight in and of itself.
Having people watch videos to enter is kind of neat if you like to make trailers, but upping your number of followers is still usually a better first choice.
Reviews Could be a Bonus...but...
Sometimes people will leave reviews for books they win, but this isn't likely. All in all, Goodreads is a better place to try if reviews are your primary goal.
Links Directly to Amazon Sites
You can add links to the messages for entering the giveaway and the win and loss sections. If you use something other than an Amazon domain, however, it won't show up as a clickable link.
I did this once for a kindle scout campaign, including links to direct people to my page, and it worked rather well. If you're promoting a book and will link directly to amazon or author central, this can be hugely useful.
People can share the giveaway through links and hashtags, and anytime people are looking at your product and seeing your name out there the better. Keep in mind, however, that Amazon has a #AmazonGiveaway hashtag that brings a lot of people who only want to click and click and won't care if they win your book. This is useful for getting people to enter your giveaway, but it will attract a lot of the wrong kind of traffic.
Giving Away Copies
An oft-ignored feature is the ability to create single-use links for your book that you can send people to give a copy. One problem with the 'gift a book' feature is that with a few clicks they can just take the money for the book instead and never actually get the book.
With single-use links you can send people copies of your eBook (for up to 30 days) that they can accept and go directly into their account. However, you cannot get refunds for books purchased through this system, so keep in mind that if you purchase a lot of 'review copy' links, the money is gone as soon as you spend it. But, if you don't use them, you can always recycle them into another giveaway in the future.
You can give books away, but you will get a lot more traction on giving away gift cards or other prizes. The thing is, you want people to buy your books, not just win a free copy, and the hope of 'winning' a copy might actually turn people away.
Big ticket products are also great for promotion, but to help with international distribution you can promote a big item, but give the prize as a gift card so they can 'buy' the item. Not all countries will have the item you are trying to promote.
Cost of Running a Giveaway
This can easily break the bank if you aren't careful. My advice would be to find other like-minded authors and work together on something like this.
My Personal Experiences
It is incredibly easy to build a giveaway and share it, though making it go viral is something else entirely. You generally need to pay to have images and more options to signup, but you can certainly run a free giveaway without much effort.
This is much like gleam.io. Both companies do a great job of giving away products and making it easy for people to enter. If you are at the stage that you want to give stuff away, then research both companies and decide if this is the route you want to go.
Don't put only half effort into it, though, because if you aren't able to gain a lot of traction it can be a costly venture with very little gain.