Whilst it is generally held that longer passwords are usually more secure than shorter ones, that is not necessarily the case. It all depends on the characters used when creating said password. Unfortunately, some websites are very insecure because they only allow alpha-numeric (and in some case only alphabetical characters (and, even more disturbingly, only lower or upper case and not both)) for their passwords.
A typical password that many people still use, despite repeated advice to use something far stronger, is password, and we can see from this that each letter is a lower case alphabetical character. That means that each letter can only be 1 of 26 different possibilities, and thus is easy to crack. Utilising both upper and lower case characters increases the security (although it is still weak) because each character can now be 1 of 52 (2*26) different possibilities, plus there is an extra factor of not knowing where upper and lower case characters appear in the password. To increase this still further, numerical characters (0-9) can be brought in, thus each character can be 1 of 62 different possibilities (2*26+10). In addition, adding special characters (such as $, %, &, etc.) can increase the security even further. In both of these latter 2 cases, again there is the additional factor of not knowing in which position a particular character type lies.
Thus we can have password (weak), passWorD (stronger, but still relatively weak), PasSw0rd (getting better), and finally, the strongest of them all, p4$sWoRd, which employs all 4 character classes. Note that these are still weak, but they just serve to illustrate the example.
For a relatively secure password (and we need to remember that it is something that we need to be able to recall, as a forgotten password is like a lock that you have lost the key for), I recommend a minimum length of 12 characters, with at least 1 character from each of the 4 character classes.
As John mentioned above, pass-phrases are also a good idea, and the security of these can also be enhanced by the same methods I described above.