Do the elliptic curves over prime fields must always contain prime number of elements (prime order)? I have gone through one example where i saw a curve defined over some prime number containing non-prime order.
In general, an elliptic curve over $\mathbb{F}_p$ can have a nonprime order. 
Much more is known.

As mentioned in Stinson's book Cryptography:Theory and Practice, 2nd Edition, Theorem 6.1, the additive group of an Elliptic curve $E$ over $\mathbb{F}_p$ with $p&gt;3,$ and prime, is isomorphic to $Z_{n_1}\times Z_{n_2}$ with $$n_2|n_1,\quad n_2|(p-1).$$