I am looking for very heavy curtains to replace these so my cat can't tear through them. I have pictures of what the hooks look like at the top and where they hook in. I need to make sure I get the right model of curtain (for a large picture window) and the right size. What would you call this kind of curtain (what would I look for on the label)? And how do I measure to make sure I get the right size? What else do I have to check to get the right ones?

 

Sorry, I somehow missed this post and I also moved it to the draperies topic.
First of all, heavy curtains won't guard against cat tears. Fabric is fabric, no matter how heavy and can be cut, ripped and teared. You might be better off getting some hard window treatments, like blinds.
If you're still interested, what you have are called "pinch pleats". So what you're looking for is pleated draperies. There are different kinds of pleats these days and any kind you like will work.
The bad news is for a large window like a picture window, it will be hard to find readymades. You may need to go custom to get the size you need. JC Penneys used to sell large sizes in readymades but I really don't know if they still do. You'll also probably need to order online if you do find any. Afraid you won't find them in stores any more.
I'm not really sure what I'm looking at in your pics, as far as the window and rod placement, but if you're using the same rod, hung in the same place, it'll probably be easiest to just measure your existing curtains. Just measure one across, as is, and that will be your finished width, then measure the length and that will be your finished length. Of course, you'll need a pair.

 

Well, if you dad them made out of canvas, that might stop a cat. Otherwise, keep their nails trimmed and maybe try some of that "keep away" type spray, but make sure you test it on some scrap material first. I would think you should be able to get a sample or piece of scrap when you shop? Really not sure about that.