It's probably the
Bot Net. That's just speculation, but it's unlikely that somebody's trying to log into your computers manually. Instead, I would think that it's an automated system that uses already compromised computers to try to compromise new systems by finding accounts with guessable passwords and logging in with ssh, in an effort to grow the ranks of the compromised servers.
I have seen these logins ever since I put my boxes on broadband. I don't worry about it because I use strong passwords and have few users. However, I would recommend you look into
Fail2Ban which uses IPTables and/or log files to temporarily or permanently firewall access from abusive IPs. I have seen it configured, and was impressed with the results.
You can also have SSH listen on a port other than 22, the standard SSH port. 222 and 223 used to be forwarded through the router to internal servers, and my logs don't show anyone unauthorized ever tried to log in to an SSH server listening on a nonstandard port (although it is technically possible to identify an SSH server running on a nonstandard port, it takes substantially longer, and therefore is probably a much less efficient attack vector than simply hitting the standard port).
I have always wanted to set up a honey pot on 22, so that I could turn around the username/password combinations I collected and use them to start fighting back against these bot nets. But I ran into difficulties getting the supplied password out of the SSH server from openSSH; I am not a skilled hacker. But I digress.
In conclusion, I do not worry about these failed logins, but some of my friends do; I usually recommend listening on a nonstandard port because fail2ban is more difficult to configure, but both are good solutions, and fail2ban requires no outside knowledge client-side (which port to connect to) and avoids possible attacks that have found your nonstandard SSH ports using smarter service identification, so it's probably a better choice all around. Of course, you can always use RSA certs rather than passwords to log in, making it absurd to try to guess the authentication info. And nothing is a replacement for strong passwords.
And remember, the originating IP of these attacks is probably not owned by an attacker, but rather to the poorly secured server of a more or less benevolent internetzien. (I like to look up the admin of the IP range if driven to action, and send them an informative email about the problem, but I have never once heard back from them. They are usually overseas).
Good luck.