Some cryptographic algorithms are as strong as the size of their key is, while other have some weaknesses that limit their strength (such as SHA-1). How strong is the ECDSA algorithm, and does that strength depend on anything (for example, the curve used and so forth)?
First of all, I'm no expert in this area. Generally $n$ bit ECC seems to have a security level of about $n/2$, but I found some claims that it's lower for certain types of curves.
RFC4492 - Elliptic Curve Cryptography (ECC) Cipher Suites contains the following table:
It doesn't seem to distinguish between different curve types.
I found an RFC draft (not a real standard RFC) that claims the following security levels:
This is consistent with other other sources that put the security level of ECC at $n/2$. Binary curves (which includes Koblitz curves as used in Bitcoin) seem to be a bit worse than prime curves (used pretty much everywhere else).
The blog entry Not every elliptic curve is the same: trough on ECC security elaborates:
For some curves (like supersingular ones) there are specific attacks, which make them significantly weaker.