I am logging on Facebook on my Android mobile but I forgot my password, so whenever I log off, I will not be able to login again. Can I see or save my Facebook password?
[Can I] save my Facebook password?
If you're using a browser on your Android OS, preferably popular ones like Firefox, Chrome, Opera etc. then you can enable Save Password functionality for websites. Mostly, it is enabled as default setting. On the other hand, if you're using an app like Facebook to log-in into your account, then it is not possible to save your password.
That said, there are some apps on Play Store where you can save your password safely, if you've issues remembering them. I'm aware of some of them, notably KeePassDroid and LastPass Password Mgr. You can look for more alternatives by typing
password manager or
password in Play Store search field.
Regarding forgot password, you can easily reset it if you've access to your recovery email (email with which you registered at FB). There are solutions available on Facebook itself for recovery. You may look at some of them here.
Note: Alagaros already mentioned about the possibility of seeing the password, so it has been avoided in this answer. :-)
No there is no way you can view your own password, once the password is entered correctly you get a session with Facebook. Your phones uses this to authenticate with Facebook. In other words, the password is no longer in the device its memory.
You can however reset it here. For this you need to have registered a phone number or email.