Not an app, but the CD-ROM series "Fritz and Chesster" is outstanding for beginners. It's been around for a while. I have several copies in a library which I lend out to kids in a chess club.
It is the only resource I know which actually teaches chess to young children who are absolute beginners without the need for much adult assistance. I would say it is suitable from age 4, despite saying 8+. It has an excellent, appealing design as well as being good on the chess side. I have not yet encountered a child who didn't like them. Some kids do find the first one too basic, and a few kids who aren't really ready for chess - like my younger dd who has learning difficulties - don't really grasp the chess ideas very well but still enjoy the games. The children I know are of a wide range of abilities.
Fritz and Chesster 1 has a cartoony storyline with exercises teaching how each piece moves and continuing to some basic concepts. It also covers ALL the rules: there aren't that many rules in chess, but most beginners don't know all of them, or they misunderstand a few. (The trouble is, most people learn to play chess from other people rather than from a proper reference source. This means mistakes proliferate. If I had a penny for every time I've heard "but my dad is really good at chess and he says..."!)
The other two in the Fritz and Chesster series teach more advanced concepts.
You can often buy the first one secondhand for about a tenner. The others are harder to find secondhand and you'll pay around £20 each to buy them new.
However, you were asking for something your son could actually play against. I have to say that the Fritz and Chesster interface for playing games against the computer is not that great.
Here's a good safe kid-friendly website for your son to play against other children: www.chesskid.com/ There aren't many free resources on the site - it's one of those frustrating sites which is constantly offering you things and then telling you that you have to pay a subscription for them - but it's good for playing anyhow.