Basically the core functionality is to track inventory, manage purchase orders in, and sales orders out. Generally that's also linked to accounting so basically as you buy and sell stuff everything is linked to your accounting ledgers.
Then there's MRP (M for manufacturing) which is usually an option - for when stuff that you're selling isn't the same as what you're buying.
So the ERP/MRP manages bills of materials for items (basically parts lists), when you want to sell something you make a sales order, it sees if you have the inventory on hand to actually make the things, and if not you can generate purchase orders, and then once you have everything it can create work orders which basically tell your factory people to go and turn the input parts into a product so you can sell it. Invoices are created from sales orders, and workers time is tracked on the purchase order, so the cost of goods and labour is automatically tracked in your financial reports as well as the revenue etc.
Not all materials are parts. For instance, if you run a soup factory you don't order 'soup parts' but bulk ingredients in the same proportions that you end up selling, but there are many other things that you need to worry about in the process of preparation and there are things that don't even make it into the end product at all that you still need during the manufacturing step (for instance, cleaning products, various chemicals). So 'basically parts lists' is a bit of a shortcut.
Yes. Most ERP's also handle 'ingredients'. Really, it is any 'inputs' to the product. And potentially, byproducts produced and also sold. All In's/Out's, and how to track them.
The M is for “material”. Material Resource Planning.