Essentials
- Dumbbells.
For most people a set of 10lbs and another set or two in smaller sizes works well. - Resistance Cables.
These go by different names, resistance bands, resistance tubing, stretch bands etc. Get Something with handles. They frequently come in packs of different resistance levels. I have seen them come in kits for about $20-$30. - A door anchor.
This goes between your door and the doorway. It lets you close the door and anchor your bands to it for horizontal exercises. Ideally, get something that goes in from the same side you are attaching the bands, otherwise, you will have to do a funky ritual of trying to hold it in place from the other side while closing it. - Some sort of object you can lean on for pushups.
I use a weight bench, but a table, stairs, desk etc will work. Something that's heavy and can't move while you're using it.
Those are the "essentials" but even most of them can be substituted for household items.
Optional
- Something to stand on.
I have a little wooden stair set I use. You can just use a step, a strong, box etc. Something that won't move. This is for knee/glute exercises. - A yoga mat.
For comfort on the ground. - A slant board.
For knee and calf exercises. I have a sturdy wooden one, but anything that will elevate your heel will do. You can use a book, or a dumbbell that won't move. - Ankle weights, ankle straps for the resistance cables, Monkey foot device.
These just add resistance to certain leg exercises. The Monkey Foot is a device that lets you grab dumbbells with your foot. The ankle straps are the cheapest. This has limited application but it's difficult to replicate this resistance without props of some sort. - A pullup bar.
You can hook the resistance cables to a high point to substitute.
The optional list items are luxuries more than anything else. Don't worry about them unless you've already got the essentials.