All registered users have a password which works like any login password. Passwords help ensure that someone does not masquerade as another editor. Editors should use a strong password to avoid being blocked for bad edits by someone who guesses or "cracks" other editors' passwords.
Some actions on Wikipedia can only be carried out by privileged editors. The most common kind of privilege is adminship. It is especially important that privileged editors have strong passwords. Administrators, bureaucrats, checkusers, stewards and oversighters discovered to have weak passwords, or their accounts having been compromised by a malicious person, may have their accounts locked and their privileges removed on grounds of site security. In certain circumstances, the revocation of privileges may be permanent. Discretion on resysopping temporarily desysopped administrators is left to the bureaucrats, provided they can determine that the administrator is back in control of the previously compromised account.
Although the definition of "strong password" is deliberately left unspecified, privileged editors are required to use strong passwords and are informed that the developers will occasionally try to crack their passwords and disable those that can be cracked.
A good example of why you should have a strong password can be read here, when several admin accounts were hacked and used for vandalism.
For advice on personal security, including passwords, see Wikipedia:Personal security practices and Keys to a Strong Password.