The IRS makes the distinction between hobby and business based on how much profit it makes. I think the cap is around $1000 or $2000 after expenses. It can bring in less than that and still be a business, but it can't bring in more than that and still be a hobby. At least, for taxation purposes. In terms of your goals and your personal life, you can call it whatever you want.
But yes, you are technically supposed to report income on anything you sell, period. In practice, most people don't do that if they're unloading old furniture on Craigslist or old records on eBay or whatever, and if you produced a clutch or two and sold a handful of babies under the radar probably no one would care (although if you got audited they might). But if you're going to be selling anything on any kind of regular basis you need to report it.
There's a variety of tax entities your business or hobby can be, and various reasons for each, such as sole proprietorship, partnership such as an LLC (even if the partnership is just you), S-corp, C-corp, etc. You should do your research or speak to an accountant if you're going into business for the long haul, but that said you can file it under self-employment income on Schedule C and it is pretty straightforward. TurboTax is your friend and will walk you through what you need to know, provided you have records of your sales and expenses. I don't know about how the rules work for breeding animals, but for manufacturing for example, there is a difference between how you handle inputs such as raw materials or pre-made components, and how you handle expenses such as tools and equipment. For animals, there's probably a distinction between feed and caging for example, but I don't know the specifics.
Different cities, counties, and states have different rules for the structure of some of these business entities, and for what kinds of license you need for various kinds of sales, and rules about things like sales tax. For example, you may need a business license to sell one type of product but not another. If you are breeding snakes, you might also want to sell some basic supplies as a retailer, in which case you'd want to find out about the rules for retail sales of pet products in addition to sales of live animals. It's maybe a minor hassle to find all that stuff out for where you live, but once you've done it you know.
Basically, as long as you have kept good records, TurboTax will tell you what to do in order to report your income from either hobbies or self-employment. As far as state and local ordinances go with regard to ball pythons specifically, you might have to make some phone calls.
Editing to add: I'm not an accountant; the above is only my recollection; I usually have to go look up all kinds of stuff all over again every time I do my taxes, and I haven't done them yet this year, so take that as you will. But I have filed lots of Schedule C's.
