You can do it on a poorer graphics card, but that's only if it has enough GPU RAM, and only if its supports a relatively recent compute version.
You _can_ run it on a CPU, and that can work fine for very simple models (like the Tensorflow demos with MNIST) but anything much larger and things really start to take forever. Like, you'll be waiting for months.
A GTX 1060 with 6 GB of RAM is a good intro card. You can find them for < $350.
The AWS P2 instances recommended for the course ($0.90/hour) have half a K80, which has access to 12 GB of VRAM. The list price for that card is $5000.
The GTX 1080 Ti has almost as much VRAM as half a K80, 11GB, and can be found for around $700.