Does the Canon Powershot A410 (Digital Camera) works only with a memory card.?

Yes - the A410 does not have any internal memory, so you would need a memory card.

you can do it without a card
but you will only get 10 pictures or less

Of course. Your question is sort of like asking if a 35mm camera only works with film in it.

While some newer point and shoot cameras come with a very small internal memory, it is more of a gimmick than anything useful. The smallest card you may want to consider would be a 1GB or 2GB memory card. Compare that to the internal memory on some cameras of 16MB or 32MB and you can see why they are more of an emergency backup than a really useful tool