How can there be insecure elliptic curves if the discrete logarithm problem is hard?
Discrete Logarithm on elliptic curves is hard in the following sense: on an $n$-bit curve, solving DL has cost $2^{n/2}$. Thus, this is infeasible only as long as $n$ is large enough to make that cost prohibitive. A 256-bit curve is large enough; a 60-bit curve is not.

Also, some curves have a special structure which can be leveraged to compute DL faster. Notably:


If the curve order is not prime or does not have a big prime factor, then DL is easier (the same applies to normal non-curve DL).
Some curves have a low embedding degree, which allows for converting DL on the curve into a comparatively easier DL in the multiplicative subgroup of a finite field extension. This is very heavy in maths. If the curve is somewhat weakened but not too much, then it can still be safe while allowing computation of a pairing, an algebraic operation which is useful for some complicated cryptographic protocols (e.g. identity-based encryption).
Some curves make it easier to implement them securely, e.g. without leaking secret information through side channels.


Generating a new curve entails trying out equation parameters and checking that the resulting curve is not one of the known “bad cases”. In practice, a non-prime order is the most frequent case; for 256-bit curves, you need to try a hundred curves or so before getting one with a prime order. Curves with a low embedding degree are extremely rare, and easy to check for.