What is the simplest way to SSH using Python?

You can code it yourself using Paramiko, as suggested above. Alternatively, you can look into Fabric, a python application for doing all the things you asked about:

Fabric is a Python library and command-line tool designed to streamline deploying applications or performing system administration tasks via the SSH protocol. It provides tools for running arbitrary shell commands (either as a normal login user, or via sudo), uploading and downloading files, and so forth.

I think this fits your needs. It is also not a large library and requires no server installation, although it does have dependencies on paramiko and pycrypt that require installation on the client.

The app used to be here. It can now be found here.

* The official, canonical repository is git.fabfile.org
* The official Github mirror is GitHub/bitprophet/fabric

There are several good articles on it, though you should be careful because it has changed in the last six months:

Deploying Django with Fabric

Tools of the Modern Python Hacker: Virtualenv, Fabric and Pip

Simple & Easy Deployment with Fabric and Virtualenv

Later: Fabric no longer requires paramiko to install:

$ pip install fabric
Downloading/unpacking fabric
  Downloading Fabric-1.4.2.tar.gz (182Kb): 182Kb downloaded
  Running setup.py egg_info for package fabric
    warning: no previously-included files matching '*' found under directory 'docs/_build'
    warning: no files found matching 'fabfile.py'
Downloading/unpacking ssh>=1.7.14 (from fabric)
  Downloading ssh-1.7.14.tar.gz (794Kb): 794Kb downloaded
  Running setup.py egg_info for package ssh
Downloading/unpacking pycrypto>=2.1,!=2.4 (from ssh>=1.7.14->fabric)
  Downloading pycrypto-2.6.tar.gz (443Kb): 443Kb downloaded
  Running setup.py egg_info for package pycrypto
Installing collected packages: fabric, ssh, pycrypto
  Running setup.py install for fabric
    warning: no previously-included files matching '*' found under directory 'docs/_build'
    warning: no files found matching 'fabfile.py'
    Installing fab script to /home/hbrown/.virtualenvs/fabric-test/bin
  Running setup.py install for ssh
  Running setup.py install for pycrypto
...
Successfully installed fabric ssh pycrypto
Cleaning up...

This is mostly cosmetic, however: ssh is a fork of paramiko, the maintainer for both libraries is the same (Jeff Forcier, also the author of Fabric), and the maintainer has plans to reunite paramiko and ssh under the name paramiko. (This correction via pbanka.)

I haven't tried it, but this pysftp module might help, which in turn uses paramiko. I believe everything is client-side.

The interesting command is probably .execute() which executes an arbitrary command on the remote machine. (The module also features .get() and .put methods which allude more to its FTP character).

UPDATE:

I've re-written the answer after the blog post I originally linked to is not available anymore. Some of the comments that refer to the old version of this answer will now look weird.

If you want to avoid any extra modules, you can use the subprocess module to run

ssh [host] [command]

and capture the output.

Try something like:

process = subprocess.Popen("ssh example.com ls", shell=True,
    stdout=subprocess.PIPE, stderr=subprocess.STDOUT)
output,stderr = process.communicate()
status = process.poll()
print output

To deal with usernames and passwords, you can use subprocess to interact with the ssh process, or you could install a public key on the server to avoid the password prompt.

I have written Python bindings for libssh2. Libssh2 is a client-side library implementing the SSH2 protocol.

import socket
import libssh2

sock = socket.socket(socket.AF_INET, socket.SOCK_STREAM)
sock.connect(('exmaple.com', 22))

session = libssh2.Session()
session.startup(sock)
session.userauth_password('john', '******')

channel = session.channel()
channel.execute('ls -l')

print channel.read(1024)

Your definition of "simplest" is important here - simple code means using a module (though "large external library" is an exaggeration).

I believe the most up-to-date (actively developed) module is paramiko. It comes with demo scripts in the download, and has detailed online API documentation. You could also try PxSSH, which is contained in pexpect. There's a short sample along with the documentation at the first link.

Again with respect to simplicity, note that good error-detection is always going to make your code look more complex, but you should be able to reuse a lot of code from the sample scripts then forget about it.