What is a good size for an image to display on a Smart Phone?

This is a very broad question. Of course, 'Smart Phone' narrows it down somewhat. However there is still a large range of phone sizes. The following are my observations and should be tested in each case.

I had a need to place couple of dozens of images in an app (generic). I was developing the app with IntelXDK using just with HTML5, actually a web app. I wanted to make sure that I could fit in the images in my app. But what size? what file size? 

I experimented with the images you can take with a smart phone (1028x768) and resized images 75%, 50% and 25%. On an emulator they look the same. I had to set the size on the IntelXDK designer so that I can see the images properly on some of smart phones.

The details are here:
This next part (Part 6) is all about placing images and text formatting:
http://hodentekmobile.blogspot.com/2016/02/develop-side-menu-app-using-intel-xdk.html
Apple iPhone 6

 Apple iPhone 5

 Motorola Razor

 Nokia Lumia 920

 Samsung Galaxy S

What is the difference between H.264/AVC and H.265/HEVC?

Both are Video Compression Standards while H.265 is more recent.

Videos are essential for enjoying the Web, YouTube, or streaming movies. The amount of data produced by any video is very large and to send it over to you over the Internet fast enough for you to enjoy is not easy. It can be sent fast by reducing its size. However, size reduction by itself is not enough since size reduction is accompanied by a reduction in video quality. A lot of processing has to be done to get efficient compression without reducing the quality.

H.264/AVC known variously as MPEG-4 Part 10 or H.264/MPEG-4 AVC, or MPEG-4/H.264 AVC is a video coding format that uses the block-oriented motion-compensation-based video compression standard. This standard was developed jointly by the ITU-T Video Coding Experts Group (VCEG) and the ISO/IEC JTC1 Moving Picture Experts Group (MPEG).

H.264 is the standard for Blu-ray Discs and all Blu-ray disc players must be able to decode H.264. Most of the following support H.264:

• Vimeo
• YouTube
• iTunes store
• Adobe Flash Player,
• Microsoft Silverlight
• And, HDTV terrestrial broadcasts.

Read more about it here.
Read about this licensing battle between Microsoft and Motorola related to H.264 here.

H.265\HEVC (High-Efficiency Video Coding) is an improvement over the previous H.264 coding technology and in some form will be the successor to compressed video. H.265 will give the same picture quality as H.264 but with better compression (giving you a smaller file size to deal with).

Video coding exploits the redundancy that exists in Videos both spatial and temporal and removes or takes this into consideration to efficiently reduce the size. This may just be using pixels, blocks of pixels, pixels that do not change at all or change very little, predicting the motion of pixels in a rapid scene change, etc.
I recommend you read this excellent paper by Elemental to understand the H.265 basics(http://go.elementaltechnologies.com/rs/elementaltechnologies/images/HEVC_Demystified.pdf).