I am looking for an app that has some of the same functionality (ssh specifically) as the Terminal application on the desktop. What different apps exist for iOS that have this functionality?
Prompt for iOS
Prompt allows you to remotely SSH into another machine. Without jailbreaking, there is no way to use Terminal to access your file system. However, I've found that if you do jailbreak your phone and want to use Prompt as a Terminal, you can set it to SSH to server
127.0.0.1 and authenticate with the standard iPhone root credentials.
Remoter: Remote Desktop (VNC) for iOS
This app does basically the same thing just a little differently. By default it allows a VNC connection for screen sharing, but it has different add ons that you can purchase that allow SSH, Telnet, and RDP.
There are plenty of SSH apps for remote connections (see Matt's answer), but it is likely impossible that a local Terminal-like app could ever exist for the iOS device itself. The three main reasons would likely be:
Numerous bash builtins and GNU/BSD utilities (ls, cp, mv, cd, even echo) would be completely disallowed because of the inability to directly access the filesystem, I believe this even extends to arbitrary access into the sandboxed directories too.
License incompatibility. This is only relevant to Bash, which is most likely the shell you use (the actual functional/interactive part of a terminal program), Bash is licensed GNU v3 which is known from previous incidents to be completely incompatible with Apple's App Store terms. Though there are numerous alternatives that appear to allow distribution/existence on the app store, for example: tcsh and zsh.
A terminal is almost worthless without 3rd party/command line applications. SSH? 3rd party binary. grep? 3rd party application. vim, svn, curl, ping, all 3rd party binaries. A terminal app would have to ship all of these, and would be incredibly difficult to ensure appropriate licensing with all of them. Not to even say how Apple would feel about so many binaries stuffed in an application bundle.
If you are willing to jailbreak, there is MobileTerminal, which together with jailbreaking, gives you many of the common GNU utilities that operate directly on the iOS device's filesystem.
It would be a fair bit or work, though, as the version of MobileTerminal available through Cydia (the jailbreak App store) is out of date, and doesn't work on many devices. You would probably have to compile + install the current version from it's sources.
iTerminal for iOS
Not an unmitigated joy, in terms of UX, but it gets the ssh / telnet job done.
OpenTerm is a sandboxed Command Line Interface for iOS
https://github.com/blinksh/blink is an SSH/MOSH client
https://github.com/ish-app/ish uses emulation to run a full Linux including any shell you want (it's slow and iOS kills it so you can't rely on it for long background commands)