1.) HTC One: Doesn't need much explanation. Great screen, speakers, performance. A well rounded phone for the begining of 2013 and even now. I'm also confident it prolonged HTC's demise a little.
2.) Galaxy S4: Great marketing and feature packed. The smartphone for the masses
3.) Moto X: Brought a smooth, iPhone like experience. Showed the world that Android doesn't need top of the line specs to perform great. Demonstrated that simple can be better. Reliable experience and a decent camera make this a great phone for the holidays. Android 4.4. already rolling out.
4.) Lumia 520/521: Hands down the best low end phone of 2013 and perhaps ever. Great performance, and decent features for a sub $100 unlocked handset. Great value and starter smartphone. Put WP8 on the map for the most part. And the App situation can only improve. If I were on an extreme budget or only needed the basics this would be the phone I would get.
5.) LG G2: Perhaps the most well rounded phone currently out. Great camera, screen, battery life and performance. LG can still put out a great product and are still relevant going into 2014.
6.) Galaxy Note 3: The most powerful smartphone out. Jam packed with features many will never use. S-pen improvements
7.) Nexus 5: Flagship specs at a midrange price. Pure Google experience and buttery smooth performance. Not as many compromises as the Nexus 4. Updates straight from Google. $349
8.) iPhone 5S: Nothing revolutionary but it took a solid smartphone and made it even better. One of the best cameras, imessage, and facetime make this a great phone for those with family invested in the Apple ecosystem. It's also a good phone by itself.
9.) Xperia Z1: Sony showed they had what it takes to make a decent flagship with the Xperia Z and made it even better with the Z1. Perhaps under looked but still a decent option with a great camera and processing package. The only drawback being the slightly washed out screen at an angle.
10.) Blacberry Z10: Controversial I know. After a long hiatus blackberry released a solid smartphone. It has all of the basics covered, and has been further refined by several OS updates. Can also run android apps. A good choice for any blackberry fans still out there. No matter how questionable their future may be.