1. What they are
Zinc carbon is the oldest chemistry still on sale. An R6 is the zinc-carbon AA and an R03 is the AAA, and they fit the same slots as alkaline cells.
2. They hold much less
An alkaline AA holds about 2700 mAh, and a zinc-carbon R6 holds around 1100 mAh. An R03 manages about 540 mAh against 1200 for an alkaline AAA, so it runs out far sooner.
3. They leak more
Zinc carbon cells are more likely to leak when they are left in a device or run flat. That crusty white mess is the electrolyte, and it corrodes the holder and the contacts around it.
4. Most batteries are alkaline now
Alkaline has become the default for AA and AAA because it lasts much longer and leaks less. Zinc carbon is still sold as a budget option, and it still works fine in a low-drain device.
5. When to choose them
In a clock, a remote or a torch used once a month, a zinc-carbon cell can be perfectly sensible. In anything you use every week, buy alkaline and pay a little more.