With su, you become another user — root by default, but potentially another user. If you say su -, your environment gets replaced with that user's login environment as well, so that what you see is indistinguishable from logging in as that user. There is no way the system can tell what you do while su'd to another user from actions by that user when they log in.
Things are very different with sudo:
Commands you run through sudo execute as the target user — root by default, but changeable with -u — but it logs the commands you run through it, tagging them with your username so blame can be assigned afterward. :)
sudo is very flexible. You can limit the commands a given user or group of users are allowed to run, for example. With su, it's all or nothing.
This feature is typically used to define roles. For instance, you could define a "backups" group allowed to run dump and tar, each of which needs root access to properly back up the system disk.
I mention this here because it means you can give someone sudo privileges without giving them sudo -s or sudo bash abilities. They have only the permissions they need to do their job, whereas with su they have run of the entire system. You have to be careful with this, though: if you give someone the ability to say sudo vi, for example, they can shell out of vi and have effectively the same power as with sudo -s.
Because it takes the sudoer's password instead of the root password, sudo isolates permission between multiple sudoers.
This solves an administrative problem with su, which is that when the root password changes, all those who had to know it to use su had to be told. sudo allows the sudoers' passwords to change independently. In fact, it is common to password-lock the root user's account, so that all sysadmin tasks have to be done via sudo. In a large organization with many trusted sudoers, this means when one of the sysadmins leaves, you don't have to change the root password and distribute it to those admins who remain.
The main difference between sudo bash and sudo -s is spoofability. sudo -s looks in trusted locations to determine which shell to execute, whereas sudo bash causes sudo to run the first bash program in the PATH, which may not be the shell you intended it to run. There could be multiple bash executables on the system, in which case you might be tricked into running the wrong one; if someone knew you had $HOME/bin in your PATH ahead of /bin and could somehow get a bash program into your $HOME/bin directory, they could cause that program to do nasty things. sudo -s effectively can't be tricked that way, since it would require that the attacker gain root access to start with, and at that point dirty tricks are no longer necessary.
However, because sudo -s gives precedence to the SHELL variable over /etc/passwd when determining the shell to use, it may still be susceptible to spoofing by another path. If someone can modify a sudoer's environment prior to a sudo -s, they could get it to run any command they desired. sudo -s could also cause a security breach less directly through other environment variables like EDITOR and PAGER.
The solution to that is sudo -i, which is a relatively recent addition to sudo. This always gives you a root login shell and resets all but a few key environment variables. Roughly speaking, sudo -i is to sudo -s as su - is to su. Because sudo -i only looks at /etc/passwd and replaces the PATH, its security is as good as the security of the root account.
You might still want to use sudo -s for those situations where you know you want to remain in the same directory you were cd'd into when you ran sudo. It's still safer to sudo -i and cd back to where you were, though.
sudo su -This way you don't need the root password, and the-makes sure the home directory is set right. – Jens Timmerman Apr 26 at 10:34