Ordinal number
The index (position) to an element in a set or sequence.
Are ordinals just indices to some table or sequence? For example lets consider the sequence $A = {2,4,7,14,25,3,1,6,3}$. If I say that I want the third ordinal of that sequence is the result then = $14$ ? Since we $a_4 = 14$ ?
Cardinal number
Is a natural number $\mathbb{N}$ that measure the size of sets (that is not well-ordered?). So If I say that the cardinal of the set $A$ is $9$, it means that there are $9$ elements of the set $A$ above?
Im just wondering if I got these definitions right. I just want a general simple explanation without many other concepts. Thanks.
You have it pretty much right. Better on cardinals than on ordinals. The number $3$ is an ordinal when you think of it in position in the sequence $1,2,3, \ldots$ and a cardinal when you say there are $3$ elements in the set $\{5,7,11\}$. You wouldn't usually say an ordinal is an index into a sequence, but your intuition is right even though that usage is not standard.